LEHIGH UNIVERSITY LIBRARIES
I1111111111111111111111111111111111111111111111111I1111111111111 .3 9151 00897680 1
Strength of Rectangular
Composite Box Girders
ANALYSIS OF COMPOSITE BOX GIRDERS.., l,.
FRITZ ENGI~JEER'NG
LABORATORY UBRARY
byY. S. Chen
B. T. Yen
Report No. 380.12 (80)
TABLE OF CONTENTS
Page
1.
2.
3.
INTRODUCTION
1.1 Background
1.2 Objective and Scope
FLEXURAL ANALYSIS
2.1 Review of Bending Theory
2.2 Elastic Moduli and Stress-Strain Relationshipsof the Deck and the Bottom Flange
2.2.1 Reinforced Concrete Deck
2.2.2 Orthotropic Bottom Flange
2.3 Differential Equations of Stress Function andSolutions
2.4 Stresses in Flanges and Webs
2.5 Results and Comparisons
TORSIONAL ANALYSIS
1
1
3
6
6
10
10
16
19
22
26
33
3.1 Torsional Sectional Properties of Single-celled Box 33
3.2' Location of Twisting Center 40
3~3 Differential Equations and Solutions 42
3.4 Results and Comparisons 46
4.
5.
DISTORTIONAL STRESSES
ELASTIC STRESSES AND DEFLECTIONS
48
55
5.1 Stresses 55
5.2 Effect of Cracks in Concrete Deck Due to Negative 58Moment
5.3 Deflections 59
6.
7.
SUMMARY AND CONCLUSIONS
ACKNOWLEDGMENTS
ii
61
63
TABLE OF CONTENTS (continued)
TABLES
FIGURES
APPENDIX A - FLEXURAL STRESSES IN SINGLE CELL
COMPOSITE BOX GIRDERS
APPENDIX B - ROTATIONS AND DERIVATIVES OF WARPING
FUNCTION
NOTATION
REFERENCES
iii
Page
64
70
121
149
154
160
1. INTRODUCTION
1.1 Background
Steel and composite steel-concrete box girders have become
increasingly popular as bridge superstructures in the last two
decades. The main reasons are that the box girders are (1) struc-
tural1y efficient because of their high torsional rigidity,
(2) aesthetically pleasing because of their long span with shallow
depth, and (3) highly economical in fabrication and in maintenance
because of their segmental type of construction and their interior
space sealed to provide a noncorrosive atmosphere.
I -1 h f f - f "1 (1.1,1.2)t was not unt~ t e our un ortunate erect10n al ures
of steel box girder bridges in Austria, the United Kingdom,
Australia, and Germany that this type of structure received
extensive research. In particular, the Merrison ~ommittee was
established by the United Kingdom Department of the Environment
to inquire into the basis of design and method of erection of steel
box girder bridges. The committee issued the "Interim Design and
Workmanship Rules,,(1.3) The content of the rules is mainly the
elastic, prebuckling stress analysis taking into consideration the
effects of shear lag, torsional warping, cross-sectional distortion,
residual stresses, and plate initia.l imperfections.
-1-
The methods of analysis and design of box girders have been
surveyed several times to date(l.4,l.5,l.6). The prismatic folded
plate theory by Goldberg and Leve(l.7) considers the box girder to
be made up of an assemblage of folded plates. This method uses
two-dimensional elasticity theory for determining membrane stresses
and classical plate theory for analyzing bending and twisting of
the component plates. The analysis is limited to straight, pris-
matic box girders composed of isotropic plates with no interior
diaphragms and with simply supported end conditions. Scordelis(l.8)
later presented a folded plate analysis for simply supported,
single-span box girder bridges with or without intermediate
diaphragms.
The thin-walled elastic beam theory developed by Vlasov(1.9)
has been refined and extended to treat simple or continuous single-
cell girders with longitudinally or transversely stiffened plate
1 d · h ··d d f bI· · d· h (1.10,1.11)e ements an Wlt r1g~ or e orma e lnterlor lap ragms .
The complexi·ty of the refined analytical methods) the "plate
element" method and the "generalized coordina~e" method, tends to
obscure the effects of the major design· parameters,. A simplified
version of the refined methods for determining the stresses induced
from cross-sectional distortion of a single-celled box girder has
been developed based on an analogy with the theory of a beam on an
1 · f d · (1.12)e ast1c oun at10n .
The finite element method is the most general of the methods
utilized. It can treat any loading and 'boundary conditions, varying
-2-
girder dimensions and material properties, and interior diaphragms.
However, more computer time is required than with the other methods.
The main problem in the finite element procedures has been to seek
a more sophisticated displacement field so that the resulting
stresses and node displacements can represent the actual conditions
more realistically. Scordelis(l.l3) has used rectangular elements
with six degrees of freedom per node in his concrete box girder
studies. Lim and Moffat(l.l4) have developed third order extensional-
flexural rectangular elements. These elements-were extensively
used in conducting parametric studies of shear lag and cross-
sectional distortion for the preparation of the Merrison Interim
Design Rules(1.3).
In addition, there are modified methods such as Finite Segment
Method(1.13) and Finite Strip Method(1.15). The finite segment
method, using matrix progression procedures, is based on simplified
folded plate theory. 'The finite strip method is the extension of
the finite element method to the case of finite strips.
1.2 Objective and Scope
Most of the methods of analysis mentioned above are primarily
used for the examination of stresses and cross-sectional proportioning
of steel or concrete box girders. The main objective of this work
is the development of a procedure for stress analysis of steel
concrete composite box girders on the basis of classical elastic
theory. From the procedure, information can be derived for pro-
portioning of box girder cross-sections.
-3-
The common configurations of composite box girder cross
sections are illustrated in Fig. 1.1. The top flange of a box
girder may be a cast-in-place or precasted reinforced concrete
slab. The bottom flange may be a plane or an orthotropic steel
plate. The webs may be plain steel plates stiffened transversely
or transversely and longitudinally. A composite section with a
single box and a concrete deck is chosen for this study. A combin
ation of single boxes form a multi-box with the webs carrying
the flexural shear and the top deck serving as the roadway. This
probably is one of the most efficient and economical arrangements
for continuous span composite steel-concrete bridges. The procedure
developed here for single cell boxes can b"e applied to multi-cell
composite box girders.
In this study a load eccentric to the shear center, as shown
in Fig. 1.2, is decomposed into bending and torsional systems.
The torsional system is further decomposed into pure torsional
and distortional systems. The bending system considering shear lag
effect is examined in Chapter 2. The concrete deck and the
bottom flange can be either isotropic or orthotropic. The
expressions for the stress distribution in and the equivalent
widths of the flanges are deduced. The pure torsional system is
considered in Chapter 3. A unified, consistent method for the'
evaluation of torsional cross-sectional properties of composite box
sections is presented. In Chapter 4 the distortional system is
discussed. The torsional and the distortional warping normal
-4-
streses are compared~ The deflection caused by cross-sectional
distortion is also discussed. In Chapter 5 the computed and
the experimental stresses and deflections are compared within the
elastic, prebuckling range. Good agreement has been observed.
-5-
2 • FLEXURAL ANALYSIS
2.1 Review of Bending Theory
The Bernoulli-Navier hypothesis states that plane cross
sections of a flexural member remain plane after bending. This
requires that the longitudinal strain of a fiber is proportional to
its distance from the neutral axis. For box girders with high
ratios of flange width to span length, shear lag effect in the
flanges cannot be ignored. This hypothesis is therefore not valid.
A more rigorous solution is required.
If the equivalent elastic constants, which are to be
discussed in Section 2.2.1, are obtained for the deck of a composite
box girder as shown in Fig. 2.1, the moment of inertia about the
centroidal principal axes of a cross section can be computed in the
same manner as for conventional composite sections. The normal
stresses are given by
Myxcr =
z I x
Mxcr = ....:Y......
z Iy
(2.1a)
(2. lb)
in which cr is the longitudinal stress, and M , M , I and I arez x y x y
the bending moments and moments of inertia of the transformed
section about the centroidal principal x- and y-axes, respectively.
-6-
The shearing stress at a point in the cross section due to· a
shear force V acting in the y-direction is given byy
,. =v Q
y xI t.
x ~
(2.2a)
duced to make the cross section of a box girder as shown in Fig.
To compute the static moment of area, Q , a cut must first be introx
2 2 d · (2. 1)• , eterm1uate. The shear flow in the cut section is
V'U' JSq =--L.
o I x 0
E.Y (E~ t i ds) =
r
v Qy xI
x(2.2b)
The compatibility condition requires that the longitudinal relative
movement at the cut be zero, thus requiring a shear flow ql'
The resultant shear flow is q,
q = qo + ql
and the resultant shearing stress is ~.
rr = .9.t.~
(2.2c)
(2.2d)
(2.2e)
substituting Eqs. 2.2b~ 2.2c and 2.2d into Eq. 2.2e~ill result in
Eq. 2,2a, with
Qx
ds§-G. t.
]. J.
Qx = ~ - § dsG. t.
1 ].
-7-
(2.2£)
In Eqs. 2.2, the symbols are:
y = y-coordinate of a point considered,
E. = Young's modulus of an individual element,1
E = reference Young's modulus used in the crossr
sectional transformation,
t. = thickness of an individual element,1
Q transformed statical moment of area of the cutx
section, about x-axis, at a point considered,
G. = shear modulus of an individual element,1
Q = adJ·usted statical moment of area for the closedx
cross section, about x-axis, at a point co~sidered,
and
§ = integral extending over the entire closed
perimeter.
If the shear modulus of reference material, G , is dividedr
through both the numerator and denominator of the fractional terms
of Eq. 2.2£, the f~11owing equation will result.
jCQxds
G.1
t.G 1
Qx Qxr
(2.2g)=
f dsG.
1t.G 1
r
~8-
G.~The term, G t i in Eq. 2.2g can be considered as a thickness trans-r
formation.
Similar to the case of Eqs. 2.2, the shearing stress at a
point in the cross section due to a shear force V acting in thex
x-direction is given by
v Qyx
'T =I t.
Y 1
and
f Qy
ds
G.--! t.G l.
rQy Q - dsy f Gi
t.G 1
r
(2.3a)
(2.3b)
It is to be noted that the equivalent longitudinal modulus of
elasticity of the deck, Et,(Section 2.2.1) should be used in the cal-
culation of the moments of inertia and the statical moments of area
in Eqs. 2.1, 2.2, and 2.3 because bending strains take place in the
longitudinal direction.
As mentioned previously, shear lag effect in the flanges must
be considered in box girders with high flange width to span ratios.
Under flexural loading, the flanges sustain in-plane shearing forces
along the lines of connection with the webs. These in-plane shearing
forces cause shearing deformations and result in nonuniform longi-
tudinal strains. This results in nonuniform stresses across the
9,,',- -
widths of the flanges. This characteristic is termed as shear lag.
To evaluate the effects of shear lag, the stress-strain relations of
the deck and the bottom flange must first be formulated.
2.2 Elastic Moduli and Stress-Strain Relationships of the Deck
and the Bottom Flange
Figure 2.3 shows reinforcing bars in a concrete deck and
stiffeners on a steel bottom flange plate which contribute respec-
tively to the overall or equivalent elastic moduli of the deck and
to the stress distribution in the bottom flange. By treating the
deck and the bottom flange of a box girder as problems of plane
stress elasticity, the stress-strain relationship for each of these
structural components may 'be established.
2.2,1 Reinforced Concrete Deck
The elastic properties of the concrete deck may be evaluated
· 1 h f ' · 1 (2.2) F'1n a manner ana ogous to t at or compos1te mater1a S . 19ure
2.4 depicts the model used by Ekvall(2.3,2.4) for a one-layer com-
posite, in which t is the thickness of the one-layer composite and
df
the fiber diameter. The matrix material is reinforced with uni~
directional, equally spaced round fibers which are securely bonded
to the matrix. Both materials are assumed to obey Hooke 1 s Law.
The modulus of elasticity in the longitudinal (xl) direction is
estimated by the classical law of mixtures.
(2.4a)
-10-
where
EI = longitudinal (xl-direction) modulus of
elasticity of the one-layer composite,
Af = ratio of the area of reinforcing fibers to the
cross-sectional area of composite,
Ef = Young's modulus of reinforcing fibers,
A = ratio of the area of matrix to the crossm
sectional area of composite, and
E = Young's modulus of the matrix.m
The Poisson's ratio of the composite is computed similarly.
(2.4b)
where
~12 = Poisson's ratio of the composite in the
longitudinal (xl) direction due to stresses
applied in the transverse (x2
) direction,
V f= Poisson's ratio of reinforcing fibers, and
\) = Poisson's ratio of the matrix.m
It is to be noted that in Eqs. 2.4 Af
+ A = 1! m
The transverse modulus of elasticity and the shear modulus
of elasticity are expressed in Refs. 2.2 and 2.3 in the form of sine
function integrals. Execution of the· integrations results in the
following:
-1tan
(2. Sa)
-11-
(Z.5b)
where
EZ
' = transverse (x2-direction) modulus of elasticity of
the one-layer composite with one-fiber diameter
thickness
G '= shear modulus corresponding to EZ',12
= the ratio of fiber diameter to fiber spacing
(center-to-center) in the transverse (x2)
direction.
aH = -.!. [.:IT _ 2
b2 2 /-b-2 - a Z
Gm
aZ = -G- ,f
Gm
b2 = 1 - G 'f
Gf
= shear modulus of reinforcing fibers, and
G = shear modulus of the matrix.m
Equations 2.4 and 2.5 are for one-layer composites with thick-
ness, t, equal to one fiber diameter, df
(Fig. 2.4). If t is larger
than df
, Eqs. 2.4 remain the same for evaluating the longitudinal
direction properties. In the transverse (x2) direction, the moduli
are again estimated by applying the law of mixtures.
-12-
d f t - dfEZ = - E ' + E
t 2 t m
andd
f t - df
G12 = t G12 '+ Gt m
(2.6a)
(2.6b)
After the evaluation of the four elastic constants, El
, EZ
'
~12 and G12 , the fifth, ~21' can be obtained by the orthotropic
identity.
The stress-strain relationship with respect to the principal
axes xl and Xz (Fig. Z.4) are as follows:
0'1 EllA EZ\)l/A a 8 1
(J = O"z = EZ\)lZ/A EZ/A 0 82
(2.8)
'T120 0 GI2 Y12
where A = 1 - \)12 \)21- The orthotropic identity, Eq.2.7, hlts
been used to make the stiffness matrix of Eq. 2.8 symmetrical.
Practically all reinforcing bars in concrete decks are either
parallel or perpendicular to the longitudinal (z) axis of the
girder. Individual layers of reinforcing bars and concrete can
be co~sidered as unidirectional composites as. shown in Fig. 2.5.
For a layer with bars parallel to the z-axis,
-·13-,
'0o
=
ifZX
crx
C1Z
cr EllA EZ\)121A 0 €Z z
(J~ cr = E2\J 12/n E2 /n 0 € (2.9a)x x
~O 0 G12 Yzx
and for a layer with bars perpendicular to the z-axis,
or in simple matrix form,
(2 .9c)
with the stiffness matrix [c] described in Eqs. 2.9a and 2.9b.
The elastic stresses of the total deck can be obtained by
staking analysis(Z.Z,Z.4) .
[ cr}n t.
= ( L: [cJ. f) . tdi=l 1 c
(2.10)
where cr and e are respectively the stresses and strains in the deck,
[c]. the stress-strain relationship matrix of layer i, t. the thick-1 ~
ness of layer i, t the total deck thickness, and n the number ofc
layers. Inversely, the strains are obtained from the stresses,
[€}n t.-1
= ( L: [c]. /-) {cr}i=l 1 c
(2.Ila)
-14-
or simply
{e:} = [sJ {a}
where...,
e: z
te} = €X
Yzx
crz
[cr} = crx
1"zx
,and
S11 812 813
[S] 821 822
823
.. 831 832
833
(2.11b)
(2. lIe)
(2.11d)
(2.11e)
It is to be noted that the continuity condition requires that the
strains in the individual layers be equal to the overall-composite
strains [e}:
[ell = [e:}2 = · · • = [E:} (2.12)
The effect of the eccentricity of reinforcing bars in each layer has
been neglected in the above derivations.
The [8] matrix of Eq. 2.11e is symmetric, with 813 = 823 = O.
From this matrix, the equivalent elastic constants of the total
deck plate are obtained.
Ez
-15-
(2.13a)
E1= --
x 822
G1= --zx 833
812
\) = -zx S11
\)xz =_ 821
822
(2.13b)
(2.13c)
(2.13d)
(2.13e)
The stress-strain relationship for the total deck is expressed con-
ventionally.
1= - (0' - \) crx)E z zx
z
1€ = - (0" - \) cr)x E x xz z
x
2.2.2 Orthotropic Bottom Flange
(2.13£)
(2.13g)
(2.13h)
For the bottom flange plate which is orthotropically rein-
forced by stiffeners (Fig. 2.3), the stress·strain relationship is
derived using the same procedures as those employed by Abdel-
S d(2.5)
aye • Two assumptions are made: (1) the shear forces are
carried by the plate only; and (2) the stiffeners are considered as
bar elements taking only axial forces. Hence, the stress-strain re-
lations for the stiffeners and the plate in the longitudinal (z)
direction are simplified.
(0") = E €Z r S Z
-16-
(2. 14a)
(2.l4b)
where
(cr ) = longitudinal stress in the reinforcingZ r
stiffeners,
(cr ) = longitudinal stress in the plate,Z p
E Young's modulus of elasticity of steel, ands
'V8
= Poisson's ratio of steel.
The axial force in a stiffener is assumed to be smeared uniformly
over the stiffener spacing, thus the equivalent longitudinal force
per unit length of the flange plate is as shown in Fig. 2.6 and
represented by
tf
(az)r E tfs
N = E € ( + ) + \) E:z s z1 -
21 -
2 s x\) s \)
s x s
Similarly,
tf
(ax)r E tfs
N = E € ( + ) + 'V e;x s x 2 s 1 -
2 s' z1 - 'V Z \)
S S
..and
N = N = G Yzx t fzx xz s
wher'e
(2.15a)
(2.15b)
(2.l5c)
N = equivalent longitudinal (z) force per unit lengthz
of plate,
N equivalent transverse (x) force per unit lengthx
of plate,
-17-
Nzx
sx
sz
Gs
= shear force per unit length of plate,
= thickness of plate,
= cross-sectional area of stiffeners in z-direction,
= cross-sectional area of stiffeners in x-direction,
= spacing of longitudinal stiffeners,
= spacing of transverse stiffeners, andEs
= shear modulus of elasticity of steel, 2(1 + ~ ).s
Solving Eqs. 2.15 for the strains results in
€ z(2.16a)
E:x
in which
(2.16b)
(2.16c)
tf
= ----2 +1 - 'J S
s z(2.16d)
=t f 'J s
21 - \) s
(2.16e)
tf
= ----2 +1 - \) s
sx
-18-
(2. 16£)
2.3 Differential Equations of Stress Function and Solutions
With elastic stress-strain relations established, the dif-
ferential equations of plane stress elasticity can be formulated to
solve for the shear lag in top deck and bottom flange. For a plane
1 (F - 2 6) h -1- b - - (2. 6)stress e ement ~g. • ,t e equ~ ~ r1um equat10ns are :
00' orr-_z+~=O'Oz OX
O'T ocr~+~=Ooz ox
and the compatibility equation is
oz OX
The stresses are related to the Airy's stress function by
(2.17a)
(2.17b)
(2.18)
crz
0X
'T zx
a2F= --oz2
o2F- - oZ oX
(2.19a)
(2.19b)
(2.19c)
Substitution of Eqs. 2.13 and 2.19 into 2.18 gives the differential
equation of stress function for the concrete deck.
(2.20a)
where
(2.20b)
-19-
If the deck is isotropically reinforced, then a
and Eq. 2.20a reduces to the conventional form.
(2.20c)
(2.20d)
c and b/a = 2,
(2.21)
For an orthotropic bottom flange, the forces per unit length
(instead of the stresses) are related to the stress function by
N -zx
(2.22a)
(2.22b)
(2.22c)
By substituting Eqs. 2.16 and 2.22 into Eq. 2.18, the differential
equation of the stress function for the bottom flange is obtained.
(2.23a)
in which
If the bottom flange has only longitudinal ribs, the (a )x r
terms in Eqs. 2.15b and 2.l6d are equal to zero. If the bottom
p = a3
(l + \) s) (a3 c - b 2)3 3
- bq =t
f3
r = c3
(2.23b)
(2.23c)
-20-
flange consists of a plane isotropic plate, Eq. 2.23a will reduce to
the conventional Eq. 2.21.
The solution to the governing differential equations, Eqs. 2.20a,
2.21 and 2.23a can be expressed as a Fourier's series(2. 7).
F =
co
1: 12
n=l Q'nX sin O! z
n n(2.24)
where annTI~' n = an integer, Xu = a function of x only, and
L = a characteristic length between two points along the girder span
at which the moments are zero. For simple beams, ~ is the span
length.
By substituting Eq. 2.24 into Eq. 2.21 for an isotropic concrete
deck, a linear ·ordinary differential equation in X is obtained, ton
which the solution is expressed as
x = A cosh Q' x + B sinh a xn n n n n
+ C x cosh a, x + D x sinh ct xn n n n
For an orthotropic concrete deck, Eq. 2.20a gives
where ,.
i i
( b + " b2
- 4ac 1/2 (b2 > 4 .)r 1 = 2a ), ac
-21-
(2.25)
(2.26a)
(2.26b)
r =2
b j 4 ., 1/2 2(- - ac) ) (b > 4ac)
2a·(2.26c)
The coefficients A , B ,C and D are to be determined by then n n n
boundary conditions at the edges of the deck at proper values of x.
Similarly, the solution for an isotropic bottom flange is:
x = E cosh a x + F sinh a xn n n n n
+ G x cosh a x + H x sinh a xn n n n
For an orthotropic bottom flange:
x = E cosh r3
O! x+F sinh r 3 Q' xn n n n n
where
9 + j 92 t
= prj 1 , 2r 3
= ( p (q > pr)
9 -J 2
=) 1 , 2r 4 = ( 9 (q > pr)
p
(2.27)
(2.28a)
(2.28b)
(2.28c)
The integration constants E through H are ~gain to be determinedn n
by the boundary conditions.
2.4 Stresses in Flanges and Webs
Because the solutions to the stress functions have been repre-
sented by the sine series, the external moment must be correspond-
ingly expressed in the same series in order to obtain the stresses
in the flanges and webs. Between two points of zero moment on a
-22-
beam or box girder, the moment can be expressed by the following
equation:co
l: Mn
sin O!n zM =x n=l(2.29a)
UTIwhere O!n = Jr' and ~ is the length between two adjacent points of
inflection or points of. zero moment.
For a simple beam subjected to a concentrated load as shown
in Fig. 2.7a,
Mn
2 P1,=
2 2n 'IT
sin Q' n 11 n = 1, 2, 3, .•• (2.29b)
and for a simple beam subjected to uniform load throughout the span
(Fig. 2. 7b) ,
Mn
4 w 1,2=
3 3n 'IT
n = 1, 3, 5, ••• (2.29c)
For continuous girders, the origin of the z-coordinate is always
selected at a point where the moment is zero. The length of the half
period, t, is terminated at an adjacent point where the moment is
also zero. To evaluate the stresses, the following general assump-
tions are made.
(a) The box girder and loading are symmetrical about
the centroidal, principal y-axis.
(b) The thickness of the concrete deck and the bottom
flange is small compared with the box girder depth. Thus
the bending stiffness of the flanges may be neglected, and
the flanges are treated as plane stress problems as done
in Sections 2.2 and 2.3.
-23-
(c) The ordinary beam theory of bending is applicable
to webs with depth-to-span ratio less than 1/4(Z.8).
(d) The existence of diaphragms is ignored in computing
the flange stresses and the equivalent widths, which are not
sensitive to the diaphragm rigidity nor greatly affected by
the presence of diaphragms(Z.8) .
(e) Complete interaction develops between the concrete
deck and the steel portion.
In addition, the following boundary conditions and strain
compatibilities are adopted:
(a) The longitudinal normal stresses and equivalent
forces are ~equal at the web to deck and web to bottom flange
junctions due to symmetry of load and cross sectiono
and
= - b /2c
(2.30a)
(2 .30b)
(b) The shearing stresses in the deck and the shearing
forces in the bottom flange are zero at the- vertical axis of
synunetry.
( rr) - 0zx x = 0 -
(N) - 0zx x = 0 -
-24-
(2 .30e)
(2.30d)
(c) The transverse displacement of t~e deck, u, is
assumed to be zero at the deck-to-web junctions.
(u)x = + b /2 = 0c
and
(2.31a)
(2.31b)
(d) The in-plane transverse normal stresses are assumed
to be zero at the bottom flange-to-web junctions by
neglecting the small protruding lips of the bottom flange as
shown in Fig. 2.8.
(Nx)x = + b /2 = 0 (2.31c)- f
(e) The normal stresses in the transverse direction and
the in-plane shearing stresses are zero at the edges of the
deck.
(,- ) -~ = 0zx x = + w /2
- c
(2.32a)
(2.32b)
(f) For simplicity, the small protruding portions of
the bottom flange outside the webs are assumed fully effec-
tive as shown in Fig. 2.9. That is, the distribution of
longitudinal forces (N ) is uniform throughout these portionsz
(Fig. 2.8).
(g) The longitudinal strains of the flanges and the webs
are equal at their junctions.
By using the above conditions and formulating the equilibrium
of externBl and internal bending moments, the coefficients A to Hn n
of Eqs. 2025 to 2.28 can be determined. The solution of these
-25-
coefficients and the subsequent substitution into the equations
for stresses are given in Appendix A. The resulting distribution
of longitudinal stresses in a cross section is sketched in Fig. 2.9.
The normal stresses are highest at the flange-to-web junctions, and
are lower in other parts of the flanges as the result of the shear
lag effect.
In practical application, an equivalent width of a flange is
often used within which the normal stress is assumed to be uniform
and equal to the maximum normal stress at the flange-to-web
junction (Fig. 2.9). The total resulting force in the equivalent
width is equal to the computed stress resultant in the flange
(Z or Z in Fig. 2.8). The equivalent widths of the concrete deckc s
and the bottom flange are listed in Appendix A.
2.5 Results and Comparisons
In order to examine the effects of deck reinforcement and of
deck orthotropic characteristics on the stresses, an arbitrary box
girder with low concrete strength and different deck reinforcement
arrangements is considered. The details of the box girder are shown
in Fig. 2.10. The girder span to box width ratio is equal to 4.
The elastic constants are:
Concrete deck:
£1 = 17.25 MN/m2 (2.5 ksi)'c
\)c = 0.17
E = 19,880 MN/m2 (2881 ksi)c
2 (1231 ksi)G 8496 MN./mc
-26-
Reinforcing bars and steel plates:
\) = 0.3s
E 203,550 MN/m2
(29,500 ksi)s
G = Gf s
78,315 MN/m2 (11,350 ksi)
Orthotropic Deck (computed for reinforcement 8/4 - 4/4):
2 (4235 ksi)E = 29,223 :MN/mz
2 (3395 ksi)E = 23,425 :M:N/mx
2(1332 ksi)G = 9,194 :M:N/mzx
\) = 0.1635, A f /A = 4.91%zx z c
\> = 0.1311, Af
/A = 1.23%xz x c
The results of the stress computations for orthotropic 8/4 -
4/4 deck are sketched in Fig. 2.11 and listed in Table 2.1. Also
listed in the table are the results for two other cases of the same
box girder geometry: one with deck reinforcing bars rotated 90
degrees so that less reinforcement is in the longitudinal (z) direc-
tion (orthotropic 4/4 - 8/4 deck), and the other with no reinforce-
ment at all (plain 0-0 deck).
The decrease in deck longitudinal reinforcement leads to a
reduction in longitudinal in-plane stiffness of the deck, hence to a
decrease of cr in the gross deck as listed in Table 2.1. However,z
the longitudinal stresses in the concrete itself increase with the
decrease in the reinforcement. The maximum stress cr in the conz
crete increases 22% from orthotropic 8/4 - 4/4 deck to plain 0-0
deck.
-27-
The influence of the 'deck properties on the longitudinal
stresses in the bottom flange is indirect. These stresses are in-
fluenced by the magnitude of the moment of inertia, I , and thex
position of the neutral axis. The decrease in reinforcement, from
a steel ratio of 4.91% to 1.23% to zero, causes only a very small
(1%) increase in the maximum longitudinal stress in the bottom flange.
For all three cases shown in Table 2.1, the effect of shear
lag on the longitudinal stresses is quite prominent, regardless of
reinforcement arrangement, as indicated by the maximum-to-average
stress ratio as well as by the longitudinal normal stress distri-
bution as shown in Fig. 2.11a.
By considering the above results, a plain concrete deck:
alone can, with due consideration of the shear lag effects, be used
for normal stress compuations for practical purposes.
The shear stress distribution is shown in Fig. 2.11b. Values
computed by considering orthotropic characteristics and by con-
sidering a plain concrete deck with either a shear lag analysis or
ordinary beam theory are all very close to each other. Therefore,
for practical purposes, a plain concrete deck can be assumed and
shearing stress computations can be carried out using only ordinary
beam theory.
To examine shear lag effects further, some experimental re
sults on aluminum beams by Tate(Z.IO) are compared with the stresses
co~puted by the theory presented herein. The dimensions of the alu-
minum alloy beams are shown in Fig. 2.12. The elastic constants are:
-28~
E = 73,140 MN/m2 (10,600 ksi)
G 27,500 MN/m2
(3,985 ksi)
\) = 0.33
Since Tate assumed no transverse displacement at the flange to web
junctions, as is assumed in this study for the top flange, the ex-
perimental results are compared with the computed top flange stresses
as shown in Fig. 2.13. Good agreement between the experimental
(dots) and computed stresses (solid lines) is evident 0 The computed
bottom flange stresses are about 5 to 6% lower than those computed
for the top flanges.
The use of an equivalent flange width to circumvent a cum-
bersome shear lag analysis has been a design practice. Equivalent
width charts are usually plotted as a function of the span to actual
· h · (2.9,2.11) f 1· (2.12)w1dt rat10 . Me fatt and Dow 1ng used rectangular
third order extensional-flexural finite elements(1.14,2.l3) to obtain
the equivalent widths. Cross section 4 as shown in Fig. 2.14 was
examined. The results from their computation and from this study are
compared in Fig. 2.15. Good agreement is observed, even in the
region of i.,/bf
< 2 (hw/~ > 1/4) where ordinary beam theory is con
sidered not applicable to the webs, but is still used in developing
the curves.
In addition to the span-to-width ratio and the loading con-
ditions, the material and geometrical properties of a cross section
also affect the equivalent width. Hildrebrand and Reissner(2.ll)
used the least work method to investigate the box beams and
-29-
concluded that the amount of shear lag depends on the G/E ratio and
the stiffness parameter m = (31 + I )/(1 + I ), where I and Iw s w s w s
are moments of inertia of webs and of flange sheets, respectively,
about the neutral axis of the box beam. The m parameter is a measure
of the stiffness of webs relative to that of flanges. Its value lies
between 1 for zero web areas, and 3 for zero flange areas. The
equivalent widths of Tate's specimen B2, meeting the conditions
m = 2 and G/E = 3/8 used in developing the curves by Hildebrand and
Reissner, are computed by this study. The two results are compared
in Fig. 2.16. Again, good agreement is observed.
It is to be noted that the computed equivalent widths of the
bottom flange, instead of the top flange, are used in Fig. 2.15 for
comparison. Since the webs of the cross section are relatively
thinner as compared to the flanges, the transverse displacement at
the web-to-f1ange junctions is considered as completely free, as is
assumed in this study for the bottom flange. For the top flange
where no transverse displacement at the web-to-flange junctions is
assumed, the computed equivalent widths are only slightly greater,
with a maximum differe~ce less than 3% from the bottom flange values.
In Fig. 2.16, for Tate's specimen B2, the equivalent width of the
top flange are plotted because the webs are relatively more stocky
and the assumption of no transverse displacement at the web-to-flange
junctions appears to be more applicable. The equivalent widths of
the bottom flange, however, are only 0.5% smaller than those for the
top flange.
-30-
Figures 2.15 and 2.16 show Winter's(2.9) curve which accounts
for the influence of the material constants (G/E), but not the geo-
metrical properties (m) of a cross section. The m values of the
cross sections in Figs. 2.15 and 2.16 are 1.15 and 2.0, respectively,
indicating that the latter cross section has relatively heavier webs.
From both figures, it can be concluded that a cross section with
heavier webs relative to the flanges has greater flange equivalent
widths, as it should be.
The equivalent widths of composite box girders with projecting
mm nnndeck portions are computed for a 3810 x 2540 (150 in. x 100 in.')
box section with the web slenderness ratio equal to 200 and the
bottom flange width to thickness ratio of 150. Table 2.2 lists the
equivalent width ratios for this cross section with deck projecting
widths w /b = 3.0 and 2.2. The latter ratio is the maximum allowedc c
by AASHTO(2.14). In both cases, the effects of span-to-actua1 width
ratio is predominant.
The influence of deck projecting widths on the equivalent widths
of the box flanges is examined in Fig. 2.17, in which the equivalent
widths are plotted as a function of the projecting widths. The near-
horizontal lines show that the influence is minor. The equivalent
widths of the projecting deck itself, on the other hand, decreases
with the increase of projecting width, as is listed in Table 2.2.
Most of the studies on shear lag are concerned with noncomposite)
stiffened or unstif£ened box sections without projecting top
£1 (2.9,2.10,2.11,2.15) w1·th hange , or concerned I-girders wit
-31-
orthotropic steel decks(2.5). This st~dy provides a procedure to
evaluate the shear lag effects on composite box sections with pro
jecting deck portions. Although for the sample composite box girders
of this study the orthotropic characteristics of the concrete deck
were found insignificant. The influence of these characteristics
may be -important for other structures such as box and TI sections
with metal-formed composite deck. These structures· can be analyzed
by using the equations developed in this chapter.
-32-
3. TORSIONAL ANALYSIS
3.1 Torsional Sectional Properties of Single-celled Box
The mathematical expressions of thin-walled elastic beam
theories for evaluating the torsional sectional properties of homo-
geneous and composite open cross sections have been developed and
·f· db· t· t (1.9,3.1,3.2,3.3,3.4,3.5,3.6)ver1 ~e y many ~nves 19a ors . For
closed cross sections composed of single material, methods for
evaluation were developed by Benscoter(3.7) and Dabrowski (3. 3) • For
composite, closed sections the evaluation of the St. Venant (uniform)
torsional constant was treated by Kollbrunner and Basler(3.l). The
non-uniform torsion of composite closed sections is considered here.
The following fundamental assumptions are made:
1. The component parts of the cross-section are thin-walled
and can be treated as membranes.
2. The cross-sectional shape is preserved by sufficiently
spaced rigid diaphragms wh~ch are free to warp out of their plane.
3. The box girder is prismatic. The thickness of the component
elements may vary along the profile of a cross section, but not along
the length of the girder.
4. The materials forming the cross section satisfy Hooke's
law. The longitudinal normal stresses will be evaluated disregarding
the Poisson's effect(1.9,3.1,3.2,3.3).
·33-
5. The connection between one material and another is mono-
lithic so that no slippage or separation will occur.
Figure 3.1 shows a thin-walled composite closed cross section.
From the uniform torsion analysis, the shear deformation in the
·ddl f d S V·· · b (3.3,3.f)m~ e sur ace ue to t. enant tors~on 18 g~ven y
where
au= OW + __s =
Yi as OZqsv
G. t.~ ~
(3.1)
Yi
= shear strain of an individual material,
w = displacement in the axial z-direction,
u = displacement in the tangential direction,s
q = St. Venant shear flow,sv
G. = elastic shear modulus of an individual material, and~
t. = thickness of an individual material.1.
For small angle of rotation and on the basis of the assumption that
cross sections maintain their shape, the quantity aU fez in Eq. 3.1s
is equal to p 0', in which p is the distance from the shear centero a
to contour tangent and 0' the angle of twist'per unit length. After
substitution of au foz the axial displacement w can be obtaineds
by integrating Eq. 3.1 with respect to s.
w = wo
s
+Jo
qsvds - 0'G. t.
1. 1.
P dso(3.2)
in which w is the longitudinal displacement at s = O. The firsto
integral in Eq. 3.2 applies only to the cell walls but not to the
open projecting parts of a cross section as shown in Fig. 3.2.
Integrating Eq. 3.2 around the closed perimeter gives rise to
the relationship
qsv I
§ ds - ~ § Po ds = 0G. t.1 1
From membrane analogy of torsion(3.1, 3.3)
(3.3)
(3.4)
in which MT is the section torque and Ao
the enclosed area of the,
closed part. The rate of twist, 0 , can be expressed as
(3.5)
where G is the elastic shear modulus of the reference material andr
KT
the St. Venant torsional constant. The second integral expression
in Eq. 3.3 can be represented by
§ Po ds = 2 Ao(3.6)
Substituting Eqs. 3.4, 3.5 and 3.6 into Eq. 3.3 and solving for KT
results in
4 A 20,
§ dsG.
1G t i
r
(3.7)
G.1The term ~ t
iin Eq. 3.7 can be considered as a transformed thick--
rness.
-35'-
By expressing qsv in terms of ~' through the combination of
Eqs. 3.4 and 3.5 and substituting it into Eq. 3.2, the following
expression for warping displacement is obtained.
,w = w -0 W
a 0
in which
s 2 A s dsJ 0 J~
= Po ds -§~ G.
0 0 ~G. ~ti~
~ti rr
(3.8a)
(3.8b)
is called the double sectorial area or the unit warping function with
respect to the shear center of the cross section. To evaluate Wo
at points in a closed cross section with open components (Fig. 3.2),
one can first pr~ce~d around the closed perimeter to find the
values for points on the perimeter by using Eq. 3.8b. This can be
executed by assuming the unit warping to be zero at an arbitrary
point of origin of integration (for example, point 1 in Fig. 3.2).
The w value for a point on the open projecting part can be obtainedo
by performing the integration, J Po ds, starting from the junction of
the open projecting element with the closed perimeter, where the wo
value has been found, ~o the point in question.
dependent on the path of integration.
Values of wareo
When the warping displacement is constrained, then normal and
corresponding shearing stresses are created in addition to the uni-
form torsional shear flow. If Poisson's effect is ignored (assumption
4), the warping normal stress is represented by
crwIt
- 0-36-
(3.9)
The total normal force due to warping on a cross section can be ob-
tained by summing the warping normal stresses over the entire cross
sectional area, A, and this is equal to zero.
N = J a- t ds = 0A
w
From Eqs. 3.9 and 3.10, a is deduced.w
If
cr = E. f/J WW 1- n
where
1 J -Wn= - (JJ dA~'c - LOoA"k 0
A
(3.10)
(3.lla)
(3.11b)
is defined as normalized unit warping with respect to the shear
center, and
E.= J dA-/e = J -1 t ds
A A Er(3.11c)
is the transformed area of the section. The values of normalized
unit warping, ill , are independent of the integration path and ren
present the warping distribution in a cross section.
In addition to the St. Venant torsional spear, the effect of
warping torsional shear on the warping displacement can be also taken
into consideration. B~nscoter(3.7) assumed that the· distribution of
warping displacement in a section is still akin to w which isn
deduced from considering only the St. Venant torsional shear.
For the span-wise distribution, a warping function f(z) is
introduced. The warping displacement is expressed as
1
W = f wn
-37-
(3.12)
The warping normal stress is obtained by
cr = E. WI = E. fll Wnw ~ ~
The bimoment is defined as
B = J cr w dAAwn
Substitution of Eq. 3.13 into 3.14 results in
B = E f" Lr W
where2
1- = J W dA*OJ A n
is called the warping moment of inertia.
(3.13)
(3.14)
(3.15a)
(3.I5b)
The equilibrium of the longitudinal forces on a differential
element (Fig. 3.1) requires that
(3.16)
where q = total shear flow, including both St. Venant and warping
shears. Substituting Eq. 3.13 into 3.16 and performing integration
leads to the expression for the shear flow.
in which
q = q - E. S _ f" Io 1.. OJ
(3.17a)
S'- =OJ
'8
Jo
(3.17b)
The sum of moment of shear flow about the shear center over the
cross section is the section torque:
M.r = qo § Po ds - Ei f"' JA
-38.;..
s~
".OJP dso
(3.18a)
from which the integration constant, q , can be determined:o
M E. f'"T 1
qo = 2A + -Z-A-- SA S(jjo 0
P dso
(3.18b)
On substitution of Eq. 3.l8b into 3.17a, it is obtained
MT
E S f " 1q = 2A - i 'Wo
-in which ~- is the warping statical moment,OJ
(3.19a)
sw = sw P ds
o(3.l9b)
The distribution of S - and of w on a cross section is dependentW n
on the dimensions of component parts. A typical distribution of
each is shown in Fig. 3.3. The total shear flow expressed
in Eq. 3.19a comprises two parts. The primary or St. Venant shear
flowMr=--
2 Ao
(3.20a)
has been given by Eq. 3.4 and occurs only in the closed part of a
box section. The secondary shear flow
E S f " ,q = - . -Vi 1 w.., (3.20b)
is induced by the warping torsion and is in self-equilibrium.
Equation 3.20b can be used to evaluate the secondary shear flow in
any part of the cross section.
The St. Venant shearing stress in the open projecting parts
of a cross section (Fig. 3.2) can be approximated by
_ M.r GiIJ.T ~ - K G t iT r
-39'-
(3.21)
with KT
value computed from Eq. 3.7. This shearing stress varies
linearly across the thickness of the projecting parts. For the walls
of the closed perimeter, ~TI is added algebraically· to the shear from
Eq. 3.20a to form the total St. Venant shearing stress.
Tsv
(3.22)
For composite steel-concrete box girders, the concrete deck is much
thicker than the steel plates and has much lower allowable shearing
h f h · d · f b f· (3 • 1)stress, t ere ore t e conS1 erat10n 0 ~T may e 0 1mportance •
From Eqs. 3.8b, 3.11b and 3.13, it can be deduced that for a
section consisting of straight plate elements with constant thickness
along the s-direction, the cr distribution is linear along the platew
profile. This implies that the warping stresses are evaluated by the
Navier hypothesis. Unlike the shear lag in the flanges of a cross
section under bending moment, the warping stresses are in effect
resisted by the plate inplane bending. Thus, as long as the width
to length ratio of the component plates is small (less than 1/2
by Ref. 3.8 or 1/4 as suggested in Ref. 2.8} ,the assumption of
linear distribution of warping normal stresses is applicable.
3.2 Location of Twisting Center
The location of the shear or twisting center is essential
for the evaluation of section properties and torsional rotations.
As depicted in Fig. 3.4, the relationship between p and Po is
P = P + y' dx _ x .s!zo a ds 0 ds
-40-
(3.23)
where p is the distance from the centroid of the cross section to the
tangent to a point P in question. Integrating Eq. 3.23 from an ar-
bitrary origin 0 to point P on the perimeter,
s s1
0Po ds = 1
0P ds + Yo x - Yo Xl - Xo Y + Xo Yl
and substituting into Eq. 3.8b gives
where2 As s
dsI p ds -0 I00= § ds G.
0 0 ~t.G. G~ ~
G t ir
r
(3.24)
(3.25a)
(3.25b)
is defined as double sectorial area or unit warping with respect to
the centroid.
By substituting Eq. 3.25a into 3.l1b and noting that x- and
y- axes are centroidal, but may not be principal axes, it is
obtained
dA*- ill - y x + x yo 0(3.26)
The w values can be calculated by either Eq. 3.13b or Eq. 3.26. Then
moments about x- and y- axes produced by the warping normal stresses
are zero, or
M = IA
cr Y t ds = 0x w
and
M = I cr x t ds = 0Y A
w
(3.27a)
(3.27b)
-41~
Equations 3.13a, 3.26, 3.27a and 3.27b are combined to give
x I - Yo I = I_0 x xy Wy
x I - Yo I = I-0 xy y Wx
(3.28a)
(3.28b)
where
I = Sy2 dA*, the moment of inertia about x-axis, (3.28c)x A
2x dA*, the moment of inertia about y-axis,I y = SA
I = Sxyxy A
dA*, the product moment of inertia,
(3.28d)
(3.28e)
SA-r- = WX dAit' the warping product of inertia,
(JJX,
and
r -r- = Wy dA*, the warping product of inertia.wY'
-'~ ....
--- .-
The location of torsion center is obtained by solving
Eqs. 3.28a and 3.28b.
I r- - I I-x = y wy xy WX
0I I I
2-x y xy
I I- I I-x WX xy LUYI I ~- I 2x y xy
3.3 Differential Equations and Solutions
(3.28£)
(3.28g)
(3.29a)
(3.29b)
The differential equations of torsion have been derived
by Benscoter(3.7) and Dabrowski(3.3). Benscoter used Galerkin's
method to solve the differential equation of displacements and
-42-
developed the differential equations relating the spanwise warping
displacements and the angle of rotation to the applied torsional
load. Dabrowski obtained the same equations by enforcing the shear
flow obtained from the equilibrium condition (Eqs. 3.18 and 3.19)
to satisfy the connectivity requirement of the closed perimeter, and
by enforcing the shear flow obtained from the displacement relation-
ships (Eqs. 3.1 and 3.12) to satisfy the equilibrium condition. The
differential equations are as follows:
E I- f'" - G K 0' = -·Mr W r T T
(6' I-L f' +M
T=
G Ir c
in which the coefficient
KT
l.1 = 1 - Ic
(3.30a)
(3.30b)
(3.30c)
is called the warping shear parameter by Dabrowski(3.3) and is a
(3. 7)measure of cross-sectional slenderness • For a' very thin
section ~ approaches unity. For a fairly thick section it lies
in the neighborhood of one half~ And
I = J P 2o
dA* (3.30d)c A 0
is the central second moment of area, or central moment of inertia.
The rate of the angle of twist, 0', can be eliminated from Eqs.
3.30a and 3.30b to obtain a differential equation in terms of
warping function f only.
Er
1OJ
f'" - II. G K f 1 = ('N r T
-43-
(3.31)
Equations 3.30b and 3.31 are applicable to longitudinal segments of a
girder subjected to concentrated torsional load as shown in Fig.
3.5. The solutions to 0 and fare
~z~ = C1 + ~(C2 cosh AZ + C3 sinh AZ) + G
rKr
(3.32a)
where
f = C4 + Cz cosh AZ + C3
sinhMT Z
AZ + G Kr T
(3.32b)
(3.32c)
Differentiating Eqs. 3.30b and 3.31 with respect to z results in the
following equations for girder segments subjected to distributed
torsional load m (Fig. 3.5).Z
m0" = ~ fIt -
Z
G Ir c
E r- fiv JJ, Gr
KT
£11 = l..L mr w Z
(3.33a)
(3.33b)
The particular solutions to ,Eqs. 3.33 depend on the loading pattern
of mz
• For 'uniformly distributed twisting moment mt
, the solutions
areIn t z2
AZ + C4 sinh. AZ)'- 2G Kr T
(3.34a)
(3.34b)
-44-
The coefficients in Eqs. 3.32 and 3.34 are determined from the
boundary conditions. At the fixed end the rotation and the warping
displacement (Eq. 3.12) are completely restrained,
0=0
and
f' = 0
(3.35a)
(3.35b) •
At simply supported and free ends the warping is entirely free.
Thus the warping normal stress is zero, and from Eq. 3.13
f" = 0 (3. 36) .
The angle of rotation is assumed to be completely restrained at a
simple support (Eq. 3.35a). At the interior support of a continu-
ous member or at a location where the applied torsional moment
changes, the continuity conditions between the left and the right
require
0L = OR (3.37a)
, ,f L = f
R(3 • 37.b)
11 11
f L = fR
(3.37c)
As examples of solution, four different cases of loading
and boundary conditions as shown in Fig. 3.5 are solved and the
results listed in Appendix B. In all four cases, the rotation of
beam section, the second derivative of longitudinal warping
function, fn, and the section torque are given. The warping tor-
sional shear flow can be obtained by taking differentiation of fIt
-45-
and using Eq. 3.20b. Illustrative plots of section torque, rotation
0, fll and ftt. are shown in Fig. 3.6. Plots of this kind covering
frequently encountered Avalues of composite box girders will facil
itate the design proc~dures.
3.4 Results and Comparisons
Test results from two model composite box girders(3.9) are
compared with computed values to examine the effectiveness of the
thin-walled torsional theory as applied to the composite closed
sections. The details of the girders are shown in Figs. 3.7 and
3.8. The loads were applied vertically (upward and downward) at
diaphragms and combined into a theoretically pure torsional load
without longitudinal bending.
The experimental shearing stresses to be discussed herein
are computed from measured strains. The shearing stresses in the
middle of webs and of bottom flanges are identified in Fig. 3.9 by
symbols. For the webs, the test results agree very well with the
computed values (by Eq. 3.19a). In the bottom flanges, however, the
computed shearing stresses are higher than the experimental values.
In Fig. 3.10, the measured rotations of the overhanging ends are
compared with the calculated results (by Eqs. B.la and B.2d) The
computations underestimate the rotations slightly. This could be
partially due to the calculation of rotations from the measured
vertical and horizontal deflections(3.9). An examination of the
effects of cross-sectional distortion is made in Chapter 4.
-46-
Overall, the proposed procedures estimate the stresses and
rotations with sufficient accuracy. The experimental results con
firm the validity of the extension of thin-walled torsional theory
to the composite closed sections.
-47-
4. DISTORTIONAL STRESSES
The thin-walled elastic beam theory as utilized in the last
chapter assumes no cross-sectional deformation. In actual box girders
only a limited number of interior diaphragms are provided to help
maintain the cross-sectional shape. Thus, distortion of cross section
may occur, particularly between diaphragms, under the distortional
load components as shown in Fig. 1.2. Figure 4.1 depicts the cross-
sectional distortion. Distortion induces, in the component plates of
the box, transverse bending moments as shown in Fig. ~4_2 and longi-
tudinal in-plane bending as depicted in Fig. 4.3, which causes
warping of the cross section.
There exist many methods of analyzing distortional stresses
of box sections. The similarity of the governing differential
equation of a single-celled box section to that of a beam on elastic
foundation was noted by Vlasov(1.9). The beam-on-elastic-foundation
(BEF) analogy was later developed by Wright, Abdel-~amad and
R b - (1.12) Th -d h hI·o 1nson . ey conS1 er t at t e tota res~stance to an ap-
plied torsional load is given by the sum o~ the box section frame
action and the longitudinal warping action. Dabrowski(3.3) also
arrived at the BEF analogy for the analysis of cross-sectional dis-
tortian of curved box girders. A· displacement method which neglected
the frame action of the box was employed by Dalton and Richmond (4. 1) .
-48-
The transfer matrix method applied to folded-plate theory was
developed by Sakai and Okumura(4.Z). A finite element procedure
using extensional-flexural elements was developed for box girder
analysis by Lim and Moffat(1.14,Z.13). The results are reported in
Ref. 4.3.
The purpose of this chapter is to reiterate some of the
results obtained on distortional analyses.
Two factors, the number and the rigidity of interior dia-
phragms, influence the distortion of a box section. It has been
shown(4.3) that, on the basis of weight, plate diaphragms are the
most efficient in reducing distortional stresses. It has also been
h (4.2) hI- 1 h- 1 b -d d ds own t at re at~ve y t ~n pates may e cons~ ere rigi as
long as no yielding or buckling .takes place. For box girders with
rigid diaphragms, distortional stresses are minimal if loads are
applied at diaphragms, and comparatively large if loads are located
between diaphragms(1.lZ,4.2) •
To examine the effects of the spacing of rigid diaphragms in
a simply supported box girder, a vertical, concentrated load which is
eccentric to the shear' center may be applied at various locations
along the span of the girder. The maximum total normal stress (sum
of the flexural, torsional and distortional parts) occurs at the
cross section under the load. The stress versus load location plots
thus are the stress envelopes. By comparing the stress envelopes of
box girders having different numbers of equally spaced interior
diaphragms, the effects of diaphragm spacing can be examined.
-49- .
Two rectangular cross sections examined for distortion are as
shown in Fig. 4.4. Normal stress envelopes for these two cross
sections h~e been computed(4.4) by a plane stress finite element
d (4.5) and are plotted in Figs. 4.5 and 4.6. The elasticproce ure
moduli of steel and concrete are 200,100 MN/m2 (29,000 ksi) -and
25,530 MN/m2 (3700 ksi), and the Poisson's ratios are 0.3 and 0.15,
respectively. For each box girder, rigid diaphragms spaced at L/2,
L/4, and L/6 are considered, with end diaphragms always present at
the supports of the 36,576 rom (1440 in.) span.
Figure 4.5 is for the box section with a width-to-depth
ratio of 2.0. Straight lines connecting the solid geometrical symbols
are the normal stress envelopes, obtained using plane stress finite
elements, for the bottom flange-to-web junction under the load
(point 3). The solid curve gives the longitudinal stresses for the
same point computed by thin-walled elastic beam theory considering
flexure and torsion, but not distortion. The distance between the
straight line envelopes and the solid curve are the normal stresses
corresponding to cross-sectional distortion. For comparison, the
distortional warping normal stresses at the mid-distances between two
adjacent diaphragms are computed by the BEF analogy and are added to
the thin-walled elastic beam theory values. The total normal
stresses are plotted using open geometrical symbols in Fig. 4.5.
These normal stresses compare very well with those of the envelopes
obtained by the finite element procedure.
-50-
Figure 4.6 shows similar stress envelopes for the box
section with a width-to-depth ratio of 1/2. Again, the total normal
stresses from the two procedures compare well. This indicates that
the classical thin-walled elastic beam theory plus the BEF analogy
can be utilized for the evaluation of total stresses in the box
girders.
From Figs. 4.5 and 4.6 it can be concluded that, regardless
of the diaphragm spacing, when loads are applied at a diaphragm, the
distortional stresses are very small. The envelopes of total
stresses are practically in contact with the solid curve obtained by
thin-walled elastic beam theory. Therefore, distortion of cross-
section need not be considered when. loads are applied at rigid
diaphragms. On the other hand, distortional stresses may be quite
high when loads are applied between diaphragms, as indicated by the
distance between the finite element stress envelopes and the solid
curves. These conclusions further confirm the characteristics
d b h( 1012,4.2)
pointe out y at ers •
For both box girders shown in Figs. 4.5 and 406, the dis-
tortional stresses are ,reduced as the number of diaphragms is in-
creased. Theoretically, these stresses can be reduced to negligible
values if more diaphragm~ are used, resulting in the solid curve
obtained by the thin-walled beam theory for point 3. However, for
these two box sections with no distortion, the highest normal stress
due to flexure and torsional warping occurs at point 4, the bottom
flange-to-web junction opposite from the concentrated load. This
-51-
stress is determined by the box girder geometry and can not be reduced
without changing the cross section dimensions. The magnitudes of the
normal stress at point 4 along the half span are plotted in Figs. 4.5
and 4.6 as dashed curves. The distance between the solid and the
dashed curves equals twice the magnitude of the torsional warping
normal stresses, and represents the influence of warping torsion on
the stresses at the two corners of the bottom flange.
Since the total normal stress, including distortional
effects, at point 3 can be reduced by adding diaphragms, it is sug
gested herein that the interior diaphragms be spaced such that the
maximum total normal stress at point 3 at mid-distance between dia
phragms is equal to or smaller than the inherent, unreducible maximum
normal stress at point 4 at midspan where a diaphragm exists. In
this way, the maximum normal stress for design is that at midspan as
computed using thin-walled elastic beam theory.
How the torsional and distortional warping normal stresses
relate to each other depends on the geometrical shape and dimensions
of the box section. In Fig. 4.5, the maximum total normal stress
including distortional· effects for diaphragms spaced at LIB (open
square) is still higher than the maximum normal stress obtained by
the thin-walled beam theory for point 4 at midspan. For the box
girder of Fig. 4.6, a diaphragm spacing of L/6 brings the two maxi
mum normal stresses to about the same level.
To examine further the effects of cross-sectional geometry
on the torsional and distortional warping normal stresses, five box
-52-
girders of the same component plate thicknesses and same span length
are studied. Two of the five have cross sections as shown in Fig.
4.4. The other three have box width-to-dBpth ratios of 5, 1, and 1/5,
respectively. The load magnitudes are such that the St. Venant
torsional shear flow is the same for all five cross-sections. The
results are listed in Table 4.1 for torsional warping normal stresses
at point 3 at midspan and for distortional warping normal stresses at
the same point under the load at the mid-distance between diaphragms.
For all five sections, the distortional stresses decrease with in
creasing number of diaphragms. For the section with bf/hw
equal to
5, the torsional and distortional warping normal stresses are of the
same sign, thus add to each other. For the remaining four sections,
the torsional and distortional warping normal stresses are of opposite
sign, and only one of the four has distortional warping normal stress
less than twice the torsional warping normal stress when diaphragm
spacing is L/6; only two of the four when L/8. These results point
out the necessity of evaluating the distortional warping normal
stresses between diaphragms after the selection o,f box section
geometry and dimensions.
The maximum transverse distortional bending stresses com
puted by the BEF analogy for the five box girders of Table 4.1 are
listed in Table 4.2. These bending stresses are for loads applied at
mid-distance between diaphragms. The magnitude of these stresses are
high when the diaphragms are far apart, but decrease rapidly with
increasing number of diaphragms. When loads are applied at
-53~
diaphragms, these stresses are practically zero. For example, for
diaphragms spaced at L/2 and loads applied at mid-span diaphragm,
the maximum transverse distortional bending stresses are 0.41, 0.35,
2and 0.21 MN/m (0.6, 0.05 and 0.03 ksi) for the box girders with
bf/hw ratios of 2, 1, and 112, respectively. Such magnitudes can
well be ignored.
The distortion of cross section also affects the box girder
deflection (Fig. 4.1). The deflection profile of the girders of
Fig. 4.5 and 4.6 are plotted in Figs. 4.7 and 4.8. Also plotted in
the figures as dashed curves are the deflection profiles from thin-
walled elastic beam theory. For both box girders, the deflections
caused by the cross-sectional distortion are greatly reduced when the
diaphragms are at a spacing of L/4. When the spacing is at L/6, the
deflection profiles including the distortional effects are practically
coincident to those by the thin-walled elastic beam theory.
In summary, the effects of cross-sectional distortional of
box girders on the stresses and deflections are relatively unimportant
l~en rigid diaphragms are closely spaced, or when loads are applied
at rigid diaphragms. Only when loads are between far-apart
diaphragms is it necessary to consider the distortional effects of
cross sections.
-54-
5. EIASTIC STRESSES AND DEFLECTIONS
5.1 Stresses
The superposition of stress and deflection induced by
flexure, torsion and cross-sectional distortion provides the total
elastic stress and deflection. The total longitudinal normal stress
at a point of a composite box girder is thus given by
(5.1)
where crB, Ow and an are bending, warping torsional, and distortional
normal stresses computed according to Chapters 2, 3, and 4, respec-
tively. The corresponding total shearing stress is the sum of f1ex-
ural, St. Venant torsional, warping torsional, and distortional
shears.
(5.2)
Four model composite box girders were tested at Fritz
E · · Lab (3. 9 , 5 •1) Th 1 f i 1ng1neer1ng oratory • e resu ts 0 exper menta
stresses were compared with computed values so as to evaluate the
validity of the theories. Two of the box girders were 3658 rom
(144 in.), and the other two 12,192 rom (480 in.) in overall length.
The details of these box girders are shown in Figs. 5.1, 5.2, 3.7
and 3.8~ The material properties are listed in Table 5.1 and the
cross-sectional properties summerized in Table 5.2. The shear lag
-55~
effect was found to be small for these box girders, therefore, the
entire cross section of each box girder was considered effective in
flexure. The effects of the reinforcing bars in the concrete deck
were found to be insignificant. The bars were, therefore, not
included in calculating the cross-sectional properties listed in
Table 5.2. In each pair of the four box girders, one (D2 and L2) had
thinner webs, thus was flexurally and torsionally weaker than the
other (DI and Ll).
In the tests of the model box girders all loads were
applied at the diaphragms. Therefore the effects of cross-sectional
distortion were theoretically negligible. For a load of 44.5 KN
(10 k) applied over a web at the midspan of girder nl, where a
diaphragm existed, the computed maximum distortional warping normal
stress and distortional transverse bending stress were only 0.676
MN/m2 (0.098 ksi) and 0.172 MN/m2(0.025 ksi), respectively, as
compared to a total longitudinal normal stress of 49.7 MN/m2(7.2 ksi~
Consequently in the computation of total normal and shearing stresses,
the distortional terms in Eqs. 5.1 and 5.2 were omitted for these
specimens. The experimental results, however, included the effects
of all the contributing factors.
The experimental stresses to be discussed in this chapter
are converted from measured strains, which are assumed as elastic
deformations. The computed and experimental normal stresses in two
cross sections are shown in Figs. 5.3 and 5.4. For the cross section
of girder Dl in Fig. 5.3, the shear lag effect was not prominent.
-56~'
The computed and tested stresses agree quite well. For the cross
section of girder D2 in Fig. 5.4, the torsional warping caused the
total longitudinal normal stress on one side of the bottom flange
to be ·46.7% higher than that on the other side. Good agreement be
tween the computed and experimental stresses is evident.
Along the length of the specimens, the comparison of stresses
are shown in Fig. 5.5 for box girder D2. In the figure stresses
were computed at the measured points and the results connected by
straight lines to form normal and shearing stress diagrams. At the
two bottom flange points. close to the webs (D2-A and D2-B) , the
computed and experimental normal stresses agreed fairly well. Along
the middle of the bottom flange (DZ-C), the flexural shear was zero,
thus the shearing stresses were due to torsional effects only. The
computed values were higher than the recorded magnitudes at some of
the points.
The validity of Eqs. 5.1 and 5.2 is dictated by the onset of
nonlinear behavior of the box girder. Figure 5.6 is a load versus
normal stress plot for two bottom flange points o'f specimen D2. The
strain gages were located in box panel 3 (Fig. 5.2). The theoretical
and experimental stresses ·increased linearly with the increase of
load. The stresses. at gage D2-A started to deviate from the pre
diction line above the buckling load of web panel 3N. The stresses
in gage D2-B started to deviate from the straight line prediction
at higher loads. Beyond the buckling load, box girder panel behavior
is nonlinear and linear-elastic predictions become invalid. The
strength of composite box girders beyond buckling load will be
discussed in a separate report.-57~
5.2 Effect of Cracks in Concrete Deck Due to Negative Moment
It has been difficult to evaluate accurately the stiffness
of reinforced concrete beams for stress and deflection calcula-
t' (5.2,5.3)~ons . In the negative moment region of composite box
girders, tensile cracks develop in the concrete deck. The stiffness
of the girder is variable along its length, being largest between
cracks where the concrete contributed to the stiffness, and smallest
directly at a crack. To facilitate the computation of stresses in
the steel plates and the deflection of the girder in the elastic,
prebucking stage, the girder stiffness may be approximated by an
average value computed using a partial deck thickness together with
the steel U-section.
An idealized schematic diagram of load-deflection relation-
ship in the negative moment region is shown in Fig. 5.7. For
initially uncracked concrete, the full deck is effective until the
development of tension cracks (OA). Then, a partial deck thickness
is assumed effective (AB). Further increase in load enlarges the
existing cracks and causes additional cracks in the deck. Hence
only the reinforcing bars plus the steel U-section remains effective
in resisting the additional load. The onset of nonlinearity (C) due
to yielding of the steel or buckling of plates may occur before or
after point B, depending on the cross-sectional proportioning of the
girder. Since reinforced concrete decks often have hairline cracks
in the negative moment region due to shrinkage and dead load, the
condition of uncracked deck (OA) seldom exists_ A partial deck
~58-
thickness can therefore be assumed initially for practical purposes.
In this case the load-deflection relationship is represented by the
line OB'C'n' .
From the tests on the four composite box girder specimens
(Figs. 3.7, 3.8, 5.1 and 5.2), it was found that in the elastic,
prebuckling stage a partial deck thickness equal to the distance
from the center of the bottom layer longitudinal reinforcing steel to
the bottom of the concrete deck can be satisfactorily considered as
effective for stress and deflection computations.
To examine this assumption of partial deck thickness, the
diagrams of load versus stress for various locations in the box girder
specimens are plotted in Figs. 5.8, 5.9, 5.10 and 5.11. Figures 5.8
and 5.9 are for total normal stresses at web points near the composite
deck, where the computed normal stresses are influenced most by the
assumption of partial deck thickness. It is seen that the partial
deck assumption gives good prediction of normal stresses for these
specimens. The similarly good prediction is observed for the
shearing stresses shown in Figs. 5.10 and 5.11.
5.3 Deflections
Similar to stresses, the deflections caused by bending,
torsion and distortion can be superimposed to obtain the total values.
Figure 5.12 compares the calculated and measured deflection profiles
of two box girder specimens. Partial deck thickness was employed in
-59-
computing the deflection for the case of negative bending. Good·
agreement is observed.
Three load versus deflection plots are given in Figs. 5.13 to
5.15. Figure 5.13 is for positive bending plus torsion with the
predicted linear-elastic deflection computed using full deck thickness.
Figures 5.14 and 5.15 are for negative bending plus torsion, for
which partial deck thickness has been used in deflection computation.
In all three cases the experimental deflections agree well with the
calculated values, confirming further the validity of the assumption
of partial deck thickness.
The good correlation between the computed and experimental
stresses and deflections indicated that the method of analysis
proposed and discussed in Chapters 2, 3 and 4, and the partial deck
thickness discussed in this chapter are valid for the composite box
girders.
-60-
6. SUMMARY AND CONCLUSIONS
This report presents a procedure of stress analysis for composite
box girders. Although the primary concern is on steel-concrete com
posite box girders, the procedure is applicable to box girders of any
materials which follow the Hooke's Law.
Within the linear-elastic range of behavior of box girders, the
shear lag effects on the flexural stresses and the warping effects on
the torsional stresses are included in the analysis procedure. The
behavior of cross-sectional deformation is discussed briefly. The
total stress or deflection at a point of a, box girder is the sum of
those due to flexure (Chapter 2 and Appendix A), torsion (Chapter 3
and Appendix n), and distortion (Chapter 4). For the cas,es examined~
the computed stresses and deflections, are in good agreement with those
from experimental studies.
From the results of this study~ a number of, conclusions can be
drawn:
1. Shear lag effects are prominent for box girders with
small span~to-width ratios. For these box girders the normal
stresses have to be evaluated by the shear lag analysis, whereas
the 'shearing stresses maybe computed by the ordinary beam
theory.
-61-
2. The heavier the webs are relative to the flanges,
the greater are the equivalent widths of the flanges.
3. The projecting widths of the deck beyond the webs
have little effect on the equivalent widths of the
flanges.
4. The thin-walled torsional theory for box sections
with homogeneous components can be applied to composite
box sections by performing proper transformations in
evaluating the torsional sectional properties.
5. The orthotropic 'properties of concrete deck due to
reinforcing bars have little effect on the flexural and
the torsional stresses.
6. The effects of cross-sectional distortion can be
reduced by adding interior diaphragms. If the loads are
applied at sufficiently rigid diaphragms, the distortional
effects are practically negligible.
7. For box girders subjected to negative bending with
or without,torsion in the elastic prebucking stages, a
partial deck thickness can be used for estimating the
stresses and deflections.
The results of flexural and torsional analyses in this study
and the conclusions above can be employed as a basis for generating
'information for box girder design. Work in this respect can and
should be carried out.
-62-
7• ACKNOWLEDG:MENTS
This study is a part of the research project "Strength of
Rectangular Composite Box Girders", sponsored by the Pennsylvania
Department of Transportation in conjunction with the Federal Highway
Administration, and conducted at Fritz Engineering Laboratory and the
Civil Engineering Department of Lehigh University, Bethlehem,
Pennsylvania. The laboratory is directed by Dr. Lynn S. Beedle and
the Chairman of the Department is Dr. David A. VanHorn.
Thanks are due the coworkers of the project, including J. A.
Corrado, R. E. McDonald, C. Yilmaz and Professor A. Ostapenko for
their assistance and suggestions. Thanks are also due Mrs. Dorothy
Fielding for typing and processing this report. Appreciation is
extended to Mr. J. M. Gera and his staff for tracing the drawings.
-63-
I· .Cf\~I
TABLE 2.1 COMPARISONS OF MAXIMUM AND AVERAGE LONGITUDINAL NORMAL STRESSES (0' )z
OF AN ILLUSTRATIVE BOX GIRDER
Orthotropi~ 8/4 - 4/4 Orthotropic 4/4 - 8/4 Plain 0 - 0 Deck
Gross In Bottom Gross In Bottom Gross BottomDeck Concrete Flange Deck Concrete Flange Deck Flange
Maximum -2.14 . -1.46 34.36 -1.903 -1.627 34.52 -1.79 34.78
2(-0.310.) (-0.211) (4.976) (-0.276) (-0.236) (5.007) (-0.257) (5.040)cr , MN/m
z
(ksi) .- -
Average -1.17 -0.81 22.43 -1.145 -0.972 22.533 -1.12 22.67
2 (-0.172) (-0.111) (3.246) (-0.166) (-0.141) (3.268) (-0.162) (3. 286)(J , :MN/m·z
(ksi)
Maximum ~ 180 . 180 . 153 166 . 166 153 159 153Average' ,0
I0\lJ1I
TABLE 2. 2 EQUIVALENT WIDTH RATIOS
w' L iJc 1 2 3 4 5 6 7 8 9 10 12.5 15 20 25-=-b' b b
f~ c
Al.307 .511 .622 .686 .7.35" .768 • 779 .819 .835 .851 .880 .900 . •927 .944
b ..'
c
3.0112 .148 .277 .401 .493 .568 .620 .639 • 700 .727 .752 .798 .830 .873 .900
w - b •. ~ ~ r ~- ~ ..... ~ ~
c c
""3 .263 .464 .580 .655 .110 .749 .773 .805 .825 .842 .874 .896 .925 .943b f
A.1 .309 .516 .631 .703 .751 .788 .815 .837 .854 .869 .895 .914 .938 .952b
c
A '2.2
2 ..251 .445 .572 .654 .709 .752 .784 .809 .829 .845 .876 .898 .925 .942
w - b;c e
A3 .263 .465 .581 .657 .711 .751 .782 .807 .827 .844 .876 .898 .926 .944-,h
f
b = bf = 3810 mm (150 in.), t = 102 (4), tf
= 25 (1), hw = 2540 (100, t = 13 (1/2),c c w
wtf = 305 (12), ttf = 25 (1), E IE = 10, E /G = 2.34, E =ft203,550 MN/m2
(29,500 ksi)sec c s
I0\0'\I,
TABLE 4.1 WARPING NORMAL STRESSES, MN/m2 (ksi) AT BOTTOM CORNER OF BOX BENEATH LOAD
hf
Torsional Distortional cr Due to Diaphragms Spaced atw
h 2 (J L/2 L/4 L/6 L/8w w
5 8.96 25.58 12.82 9.10 6.07
(1.30) (3.71) (1.86) (1.32) (0.88)
2-6.48 54.95 27.51 19.65 14.13
(-0.94) (7.97) (3.99) (2.85) (2.05)
1-16.14 84.60 50.75 33.85 25.37
(- 2.34) (12.27) (7.36) (4. 91) (3.68)
1/2-22.-76 82.33 41.16 29.44 21.17
(- 3.30) (11.94) (5.97) (4.27) (3.07)
1/5 -28.54 58.06 29.86 22.27 14.89
(- 4.14) (8.42) (4.33) (3.23) (2.16)
TABLE 4.2
TRANSVERSE DISTORTIONAL BENDING STRESSES, MN/m2(ksi)
AT THE BOTTOM OF WEBS
bfDiaphragms Spaced at
LIZ L/4 L/6 L/Bhw
12.14 (1.45) 0.41 0.145 (1.76) (0.21) (0.06) (0.02)
2 59.16 8.89 (2.41) 1.03
(8.58) (1.29) (0.35) (0.15)
188.30 28.26 8.96 3.791 (27.31) (4.10) (1.30) (0.55)
52.82 7.93 (2.14) 0.901/2 (7.66) (1.15) (0.31) (0.13)
8.83 1.03 0.28 0.071/5 (1.28) (0.15) (0.04) (0.01)
-67-
TABLE 5.1 MA.TERIAL PROPERTIES OF SPECIMENS
(All stresses in MN/m2
(ksi)
Properties ID1
II D2 Ll L2
Steel
Small Top Flanges
YieldWebs
Stress*
Bottom Flange
213.9
(31.0)
207.0
(30.0)
253.2(36. 7)
293.9(42.6)
262.2(38.0)
253.2J (36. 7) I
r- 392.~ !(56.9) t
I 259.41(3706)
I0'\ex>I
Young's Modulus of Elasticity
Shear Modulus of Elasticity
203,550(29,500)
78,315(11,350)
Ii·lIConcrete
Deck
Poisson's Ratio
Compressive Strength*
Young's Modulus of Elasticity I
Shear Modulus of Elasticity
34.5(5eD)"'
25,530-(3700)
10,902(1580)
38.0(5.--5)
25,806(3740)
11,040(1600)
0.3
38.0(5.5)
27,462(3980)
II 11, 730( 1700)
29.1(4.21)
23,391(3390)
10,005(1450)
Pais'son "s Ratio 0.171
Yield Stress of Deck Reinforcement* I 483 I 331(70) --J (48).... - - - .---------..:;...".------
TABLE 5.2 SECTION PROPERTIES ,OF SPECIMENS
Specimens
Properties Dl D2 Ll L2
Transformed Area 106.13 104.26 554.77 492.26of Section,
2 (. 2) (16.45) (16.16) (85.9.9) (76.30)cm 1n.
Distance from N.A. 25.27 25.55 74.88 73.51to mid-line ofBottom Flange, (9.95) (10.06) (29.48) (28.94)
cm (in.)
Moment of Inertia 4 4 1.084 x 106 62.097 x 10 2.061 x 10 1.006 x 10about x-axis, I ,
KZ.603 x 104) (2.416 x 104)4 4 x (503.8) (495.2)em (in.)
Moment of Inertia 4 4 1.235 x 106 ' 6
4.87'8 x 10 4.828 x 10 1.074 x 10about y-axis, I ,
(2.966 x 104
) (2.580 x 104 )4 4 y (1171.9) (1160.0)em (in.)
- .~-
~-:-.
Shear Center 2.512 2.845 3.647 3.472above N.A. ,
(0.989) (1.120) (1.436) (1.367)em (in.)
St. Venant 4 4 8.171 x 105 6.918 x 1051.425 x 10 1.206 x 10
Torsional 4 104 )Cons tant KT, (342.3) (289. 7) ~1.963 x ·10 ) (1.662 x
4 (. 4)em 1n.
Warping Moment of 6 610
8 108Inertia, I w 1.588 x 10 1.862 x 10 3.555 x 4.090 x
6 (. 6) (5915.2) (6934.5) ~1.324 x 10 6) (1.523 x 106)em l.n.
Central Moment 4 4 1.170 10 6 6of Inertia, I ,
2.351 x 10 2.277 x 10 x 1.084 x 10
4 . 4 e (564.8) (547.0) '2.810 x 104) (2. 605 x 104)em (~n.
Warping Shear 0.3940 0.4704 0.3017 003619Parameter, ~
-69-
1---1Single Box
Multi-box
\~l 1
1 1 \ZS/
CD 7Multi-cell
Fig. 1.1 Types of Box Girder Cross Section
-70-
TABLE 5.2 SECTION PROPERTIES OF SPECIMENS
Specimens
PropertiesD1 DZ L1 L2
Transformed Area 106.13 104.26 554.77 492.26of Section,
2 (. 2) (16.45) (16.16) (85.99) (76.30)em ~n.
Distance from N.A. 25.27 25.55 74.88 73.51to mid-line ofBottom Flange, (9.95) (10.06) (29.48) (28.94)
em (in.)
Moment of Inertia 4 4 1.084 x 106 62.097 x 10 2.061 x 10 1.006 x 10about x-axis, I ,
K2.603 x 104) (2.416 x 104)4 4 x (503.8) (495.2)em (ino)
Moment of Inertia 4 4 6 64.878 x 10 4.828 x 10 1.235 x 10 1.074 x 10
about y-axis, I ,(1171.9) K2.966 x 10
4) (2.580 x 104)4 4 y (1160.0)
em (ino)" -...~ ~
f---:' •.
Shear Center 2.512 2.845 3.647 3.472above N.A. ,
(0.989) (1.120) (1.436) (1.367)em (in.)
St. Venant 4 4 8.171 x 105 6.918 x 1051.425 x 10 1.206 x 10
Torsionalx 104) 104)Cons tant K
T, (342.3) (289 •7) ~1.963 (1.662 x
4 (. 4)em 1.n.
Warping Moment of 6 10610
8 108Inertia, 1w, 1.588 x 10 1.862 x 3.555 x 4.090 x
6 (. 6) (5915.2) (6934.5) ~1.324 x 10 6) (1.523 x 106
)em ~n.
Central Moment 4 4 1.170 x 10 6 6of Inertia, I ,
2.351 x 10 2.277 x 10 1.084 x 10
4 . 4 c (564.8) (547.0) 12.810 x 104) (2.605 x 104)em (1n.
Warping Shear 0.3940 0.4704 0.3017 003619Parameter, J.L
-69-
1---1Single Box
l __n 1Multi-box
\-----../
\.......--..-/ /
V'SJ
'CD 7Multi-cell
Fig. 1.1 Types of Box Girder Cross Section
-70- ~
P P/2 p/2 p/2 tP/ 2,r ,r
",r
~ l
h - +-,r---I. b --IFlexure Torsion
p/2 tP
/2Pb Pb- -
" ... 4h ... 4h
=P~t (T=~ tP~+~· I tP/4
.. Pb .. Pb-4h 4hTorsion Pure Torsion ,Distortion
Fig. 1.2 Decomposition of Box Girder Loading
-71-
ttf L
Fig. 2.1 Single Celled Composite.Box Girder
-
-72-
- ~2 f/2
(a) Box Girder
(b) Shear Flow
Fig. 2.2 Flexural Shear in a Single Celled Box
-73-
zBars
Concrete Deck
y
t J...~~~--""'-..,aA-~fTCo z )r
.' Orthotropic Bottom Flange
Fig. 2.3 Config~ration of Deck and Bottom Flange
of.Composite .Box Girder
-74-
,,X2
Plane
Matrix
Section A-A
Fig. 2.4 Unidirectionally Fiber-Reinforced Composite
-75~
+-+-+-+1--1--1-++- -t- -I-++- 4- -1-+ .. z+--+--1-++--+ I ++-+-1-+
tXI
I II II I x2 ZI II I ----I II II I
~ X
III,II 'II
z-Bars
~~~~H;-i ~ ;;- i ~ ~~~~:A-A
(a )R.C. Deck
. ~ x2
~ X (b) Unidirectional Layers
Fig. 2.5 Reinforced Concrete Deck and Unidirectional Layers
-76~'
OCTxCTX + AX dx
(lNx(Nx+ ax dx)
OTXZTXZ + ax dx
aN(N + xz dx)
xz oX
OfTzCTZ + aZ dz
cJNz(Nz + az dz)
dx
aNzx(Nzx+ az dz)
T + OTZX dzZX oz _------....
I1I
Io-x---..... ,'
(Nx) "T
XZ"(Nxz) ti-'-------..
'zx(Nzx)
z
Y.
......-.-x
Fig. 2.6 Plane Stresses(Forces) on an Element
-77--
----_!_p-----.,d(i _I _-a' _~
-1--:=- Z
(0)
M (1)={£:Sjn anr Sin;n=1 n.".
wgllllllll;llllllllllllll~
_. i· _
~ z( b)
~MX~to 4w n2
M () ~ Jt., S' n1TZx 1 = L., 33 In.t
n=1 3 5 n -rrt 1 .
Fig. 2.7 Mbment Diagrams of a Simple Beam
-78-
<t /.,Sym.
././. Concrete
/Zs
ZeMx ............-..t'----~.
C.G. OfBoxSection Y
Fig. 2.8 Components of a Box section and NormalStresses and Forces
-, ,...--~
[~ Ita-z! ~
x...-reGe
y
]
__ __ .J
I.~JL _
~
Fig. 2.9 Longitudinal Normal Stress Pattern andEquivalen~ Widths of a Box Section
-80-.
445 KN (lOOK)
rI?n...---.---------------~--.........-----;;9,-.~.z 2540mm ..I~ (100")
<t
\,
229 mm, !_..I----(2-80-03-,~)-m-m-- ..-I.. 10 16 mm..t 10wmm(911) ~ 1(4").-,-------........._----_...-~-----------.I-
16~(5/8~')
h.......
x~
, 5mm ---......::... 1016 mmy (3/16") 1~
=========~:!:I- .....1016 mm 1092mm ~omm
......-----t~---___.. (3/8")(40") (43")
... ,...
#8(0) 102mm (4"Hz)
Deck Reinforcement( Orthotropic 8/4- 4/4)
Fig. 2.10 An. Illustrative Box Girder
-81-
o a.45MN/m2 DeckI I
34.5 MN/m2 Steel(5 ksl)
+
(a) Longitudinal Normal Stress Distribution
Il
1,11
(b) Shear Stress Distribution
Fig. 2.11 Stresses at Midspan of the Illistrative BoxGirder (Orthotropic 8/4 - 4/4 Deck)
-82~
Plate Diaphragm
2134mm (7'-0") B2
C3x 1.45 81C4xl.88 82
1524mm (5'-0") BI25mm
(I")
·E.E ......- -~==-=-===:-::-:===t:~~==:-:-:===-~=.o~ - -m N r-----T-----T--.~ ~ I
~~; :E ~ I I....... L 1. .1.__ __J. J. .L.
rt> -"'---111'0+--- - - - - -
,10
Plane
Elevation
]76 mm (3") BI
I I02mm(4",B2......-.............J59mm 153mm
I mm (0.04 11) BI aB2-........
,( Top a Bottom)
Cross-Section
Fig. 2.12 Tate's Specimens Bl and B2 (Ref. 2.10)
-83- -
13.4KN 81 17.SKN 82/\.--__(_3_K_) i (4_
K_)__-..
d?»? I ,.£.1~...__7_62_m_m~~.... I067mm'-4
•
•(15)
(10)
OZ (5) OZ·
0
MN/m2 (ksl)
BI 82
Fig. 2.13 Flange Stresses at Midspan of Tate'sSpecimens Bl and B2 (Ref. 2.10)
-84-
r-------------------.--i25 mm
1803 mm(71")
25mm'13~1~~~~~3~6~45~m~m~~~~~I~. (I")
(143.5") 13mm(0.5 11
)
Fig. 2.14 Cross Section 4 of Moffat and Dowling(Ref. 2.12)
-85-
0.2 Al/ !I.. lal
ct.
--This Study
• Moffatt a Dowling
----- Winter0.4
0.8
A3
bf0.6
o 16
Fig. 2.15 Comparison of Equivalent Widths of Cross Section4 from MOffat and Dowling (Ref. 2.12)
-86-
-- This Study
---- Hildebrand aReissner
_.. - ..-- Winter
0.8
i0.2 7J9n11/2Jm=2G/E =3/8 et
0 2 4 6 8 10 12 14 16lIb
c
0.4
Fig. 2.16 Comparison of Equivalent Widths of Tate's SpecimenB2 from Hildebrand and Reissner (Ref. 2.9)
-87-
0.8
Alb
0.7
,- We -I A3/bf'
i/be=i/ bf = 5
~
0 0.4 0.8 J.2 1.6 2.0(we-be)/be
Fig. 2.17 Influence of Deck Projecting Widths on theEquivalent Widths
-88-
C: Centroid
SC: Shear Center
z
y.
(a) Composite Closed Cross-Section
s
Tangent
(b) Mid-Line Contour y
q...
...q + :~ ds
(c) Differential Element
Fig. 3.1 Thin-wa11Bd Composite Closed Cross Section
-89-
(0) Pure Torsional Shear
1 - ~ 6 .. 12
8 14 5 II
~.... .... -9 :3 4 10
(b) Integration Paths
Fig. 3.2 Composite Closed Cross Section with OpenProjecting Elements
-90-
W n-Diagram
Fig. 3.3
Sw - Diagram
Typical Distributi~ns of Wn and S on aCross Section W
-91-
x-~-+-----~--...a
y
Fig. 3.4 Geometrical Relationships Between p and po
-92-
·~lTC
779n-(a ) f=fL ~~ (I-a) L
--IA B GI(Tnf>Q J7977
z L(b)L1
(c)
mz=mtJvcCCCCCC B
jE aL ...1... (1-a )L
(d )
;lcccccccC'
mz =mtL
B.C.ls: A 8 C CI..
~=O <PL =~R ~=O
f": 0 I I fll= 0 f": 0fL=fR ·
f" - fllL - R
Fig. 3.5 Illustrative Loading and Boundary Conditions
-93-
~
Section Torque +
Section Torque
fit
+
T
Fig. 3.6 Typical Diagrams of Section Torque, 0, £11 and f"·
-94-
2 13 141516 E13114115 116
16x51 (5/aX2) Outside(Typ~ LI a L2)
2-16x64( 5/8 X 2 Yz )
71 8 I 9 I 10 I II 112
127
635
4572
(180)
4 (ii)762 (30)
Plate Diaphragm(Typ. LI a L2)
w
127(5)
Girder LI
E
127
9
3048
8
2 (d) 1397
7
102 x 1829 (4 x72) (Typ. LI a L2)K-Frame
4
(40) I (45)
4572
1-16x64
10161 2-@ 1143 11016
127
3
1397
2
1524
(60)4572
(55)
1397
ct. Span
; J 16 x 51 Inside a Outside(Typ. LI a L2)
wGirder L2
1\0U1I
Fig. 3.7 Specimens Ll and L2 (All Units in rum (in.)
1829 (72)
N1I~16x51
'*---- Lt
381(15 )
s
1016(40)
10 -r- 1 151(0.38) T--151 1... ... r-- Ali Units In mm (in.)
(2)
51 x 13 ep@ 178( 2 X 1/2 ep @) 7) I -.l
..l- . - ::r>-13 (1/2) Cleari • -,
I 305 I 16X229(0.6IX9)... (12) >-lr 1 16 (s/a) Plate Diaphragm
7 x 1016 (0.26 x 40) LI
5 x 1016 (0.21 x 40) L2I~(J\
I
Fig. 3.8 Cross Section at Plate Diaphragms of Ll and L2
A () B
c
I.JT~y
9589mm (377.5") LIz =
9970mm {392.5")L2
(4000)400
TKN-m
200 (2000)
(kip-in.)
. 0
ABCLI 0 6. c
L2 CD A II
Webs
(3) (4)
10 20
T MN/.m ksi, ,30
(4000)
400 D
L2 LI
~T
KN..;,m200 (2000)
(kip - in.)
o
•
10T MN/m2
(3) (4)
20ksi
30
Fig. 3.9 Shearing Stresses in Specimens Ll and L2
-97-
(4000)
400
T'
KN- m200
(kip-in.)
o 0.002
ep ,radian
epLI 0
L2 G
0.004
(14)
400
200 (2)
o 0.002
ep , radian
0.004
,Fig. 3.10 Torque versus Angle of Rotation
-98-
IIII
_____ --1
Fig. 4.1 Cross-sectional Distortion
Fig. 4.2 Diagram of Transverse Bending Moment Due
to Cross-sectional Distortion
-99-
(a) Longitudinal Normal Stresses
( b) Shearing Stresses
Fig. 4.3 Distribution of Distortional Warping Stresses
-100-
2438 mm(96")
mm( I")
1'IIII(4s"tlI, b" ~:,CIIl:'OQ,"O::b' ,', '~".~'.,>:.'~ .··:0·.··0· ::1
Jl
late II
> hw=
3mm too-. -.... -1/2") I
"I -- 25II.. ~ T
bf = 1219 mm
bf / hw =1/2
I(
51mm P(2
11)
1219mm
L203mm
(Sll) -r
r: 1>:. '.4 ". '(:)', '.' .~:~ I. '"~... () '.' :<1:".!\'. '.'A.:Q : '.':~ ' .. ~: .Q'.' :'0':1j~
51 I 1219
13 - ~
l'- - 251_ _It2438 mm
(96 It)
Fig. 4.4 Cross Sections Examined for Distortion
-101-
p
i(8)
40
:MN/m2
(ksi)Diaphragm
160 Spacing BEF FEL/2 0 •Lt4 A at.
(20)LIs v 'tV
120Lta 0
0
O""z ( 16).)
@).~
(12) ~®
80
Fig. 4.5 Stress Envelope for Cross Section with bf/hw = 2
o 0.1 0.2 0_3LOAD LOCATION
0.4 0.5
-102-
p
1# i ¥
I~ L= 36576mm ..!P=1704kN (383k)
(12)
( 16)
40
80
120
:MN'/m2
(ksi)
(24)0
160
@)
(20)-
CTZ
Fig. 4.6 Stress Envelope for Cross Section with bf/hw = 1/2
o 0.1 0.2 0.3
LOAD LOCATION
0.4 0.5
-103-
Diaphragm
5(2.0) Spacing at
L/2L/4
4( 1.5)
8em 3
(in)P
( 1.0) ~ ¥
2 t... L=36576mm "'1
(O.5)P=1780kN (400k)
o 0.1 0.2 0.3LOAD LOCATION
0.4 0.5
Fig. 4.7 Deflection Profile for Bottom Flange Corner 3,
b/hw
= 1/2
-104'-
8em. (i n.)
12.5 5DiaphragmSpa.cing .at
L/2
10.0 4 L/4
7.5 3p
~t ;}b
5.0 2 .I ...L= 36576m~
P=1780kN2.5
@
... -4
0 0.1 0.2 0.3 0.4 0.5LOAD LOCATION
Fig. 4.8 Deflection Profile for Bottom Flange Corner 3,
bf/hw = 2
-105-
(24)
15 I 1611 I E
2(Q)~02...1:@)20~~02
(2(Q)8)
610
5 x25 ( 3~6 x I ) Outside(Typ. DI a D2)
10111112113114
----2--5 x 32( 3~6 x 11/4 )
213141516
(60) ~
7 @> 71J2 =521J21524
Plate Diaphragm(Typ. 01 a 02)
~ ~ 33/4) 7 @) 191
wGirder DI
62 x 838 (2 7"'6 x 33 )( Typ. DI a D2 )'--K-Frame
10 x 25 (3/ax I) Outside( Typ. 0 I a 02)
2 I 3 1111 4 I· 5 I: 6 I_~_J_Jll __~_ II IE" I
1-5 x 3·2 #In:11O~3 @ 457 =1371 381 305 381 508
(3 @ 18 =54) 51 ( 15) (20)(2)
1524 -+- 1524 -+- 610(60) (24)
Fig. 5.1 Specimens D1 and D2 (all units in rom (in.)
wGirder 02
It---to0\I
838 (33)
25x6ep@I02
(I X 1/4 ep ® 4) .. ··v·· .. ' ·~··1*-6(1' )CI -·'.1. '-:'0'· .:.~ .b:··.":·;: «= '4 ear
, II I > 5 x76 (3~6'x3)
, II I 5 (3IiS) Plate
1.9 x305 (0.076 x 12) 01111M
).6 x 305 (O.06) x 12) 02
lOx 25 (3/8 x I >
127
,.. 102 "I
!;>ID~ i i
6ep x 6ep x 102 x 102
(~32 ep x""32 ep X 4X 4) Mesh
305(12)
62(271r6)r~5(31,6) ·
II--lo"-JI
(2Y2)1 (5 )216
ct.(8 1/2 )
Fig. 502 Cross Section at Plate Diaphragms of Dl and D2
(All units in rom (in.))
3.45MNfm2
I~·5kSi)
•
--Theory
TestI.Jij524m~1
(60")ct ~
lp! t[-~-(-)---
P=80.1 kN (18k)
tl
(+)
I J
o 345MN/m2
(5ksi)Steel
Fig. 5.3 Normal Stresses in the Cross Section of
Girder D1 at Z = 1429 rom (56.25 in.)
-108-
3.45MN/m2
I~o.5kSj)
•
=
-- Theory
Test
ctI . IP1191mm
I(7.5 11)"t(- )
[ ;
P=80.l kN (18k)
tl
(+ )
I Io 34.5MN/m2
(5 ksi)Steel
Fig. 5.4 Normal Stresses in the Cross Section of Girder
D2 at Z = 1397 rom (55 in.)
-109~ .
-H-25- (I )
02-A
D2-C
191(7.5)
2756(108.5)
2070(81.5)
I
Tension
P=80.l kN·(18k)
~I.fJ>
t-Z
z =762mm 139769MN/m2 (30
11) (55)
(lOks;) I I
o
02-8
•
---Theory
Test
Tension
69MN/m2
(10ksi)
o
Shear
6.9MN/m2or (I ksi) + 02-C
Fig. 5.5 Distribution of Normal and Shearing Stresses in theBottom Flange Along the Span of Girder D2
-110-
100 150(J, MN/m2 (ksi)
250
(35)(30)
200
(25)(20)(15)(10)
50
( 5)
o
Fig. 5.6 Load versus Normal Stresses in the Bottom Flange of Girder D2 at Z = 1397 mm (55 in.)
By' Reinforcing Bars +Steel U-Shape
LOAD
o
Onset OfNonlinearity
Cracking Load
Deck
DEFLECTION
--&-
Fig. 5.7 Schematic Diagram of Load-Deflection Relationship
in the Negative MOment Region
-112-
PKN (kips)
175150
Reinf.Only
~
",I!I''/ Test
75 100 125cr, MN/m2 (ksi)
Portia IDeck
50·25
FullDeck
75
(15) -, ~prIfT ;;g; ,t=z 3048 mm . .:Gd
50 (120") (24)
P
25 t25 (I) J N
DI-A
o
(20)
(25)
100
Fig. 5.8 Load versus Normal Stress in the Web of Girder Dl
at Z = 3099 rom (122 in.)
-113-
PKN (kips)
100Partial Re'inf.
(20) Deck OnlyD2-E 02-D D2-E D2-0
75D'2.~~"""
..... .-- ~ Test.-A
(15) .,.,.. ----___ ---\)'2....D"".,."'".",.
lPI
50 25 L( I) -r:2-E 02-0
25 J;-JP~
t7"Z 3048 mm ~I~(15) (20)
.~. , 0 25 50 75 100 1250-, MN/m2 (ksi)
Fig. 5.9 Load versus Normal Stress in the Webs of GirderD2 at Z = 3099 rom (122 in.)
-114-
15050 75 100 125T, MN/m 2 (ksi)
25o
PKN (kips)
150 Theory, Partial Deck
--- Test(30)
125
(25)
.".".
100
(20)
15N
75
t( 15)
~ ;79;'
C Z3048 mm -~50
p
l25 152~ s N 01-8(6)
1DI-C
Fig. 5.10 Load versus Shearing Stresses in the Webs of Girder
Dl at Z = 3099 rom (122 in.)
-115-
15012550 75 100T, MN/m2 (ksi)
25o
(30)Theory, Partial D,eck
--- - Test125
.".".""".
"DI-E \ .;'
(25) .,JP/
'"100
;"
/p I.
KN (20) /
(kips)Per,15 N
75 fII"I""'''''''01-0 ,..( 15)
~
\'"fill"
........ 4
50~
I .p(10)
;g;.J~z 3048 mm
~p
25 152~(5) (6)S N
1: 01-0
DI-E
Figo 5.11 Load versus Shearing Stresses in the Webs of
Girder Dl at Z = 3353 rom (132 in.)
-116-
p
.l,a--I~
P=53.4 KN (12 k)
iD2dj;TI .. 1524 mm
(60" )
P=66.8 KN (J 5 k) p. ,~S;;»;;; ,
.. 11III61oJ(24)
3048(120)
2.5 mm(0. JIt)
o~-~--------------:l'I!d
01-d£TI ..
Pa rti aIDeck
Fig. 5.12 Deflections along the Span of Box Girders Dl and D2
-117-
8
p
1... 1524mm ..I~z
Per, IN
~-"'-7···C................,...--(50) ".
,Ir . . Keo .... .. . PuIt (Test ) = 237. 6 N
~ K...... -A' (53.4 )
r"r(
Ift~ p,
50
(40)
100
250
350
300
(30)
200P
KN( kips)
150
30 40 508 mm (in.)
o
(0.5)
10 20
(1.0) (1.5) (2.0)
60 70
Fig. 5.13 Load versus Deflection at Midspan of Girder D2
-118-
PkN (kips)
8
p
.;;s;
~z
3048mm------__��D........--raa-t
/1~~,/ ,/ '/ ",
Partial /e'" Pult(test)=1I1.7kN .......Deck fIJI' (25.lk)
/'<sttI
II,fj,
I,"1
(25)
75
50
100
. 25
(0.5) (1.0)
o 5 10 15
8, mm (in)
20 25
Fig. 5.14 Load versus Deflection at Overhanging End of
Girder D2-i19-
1000 Full Portia I ,IJDeck Deck ""(200) , Test
Per, 13 N/
800 II
I/
(150) I
600V
p ~
'IKN p
(ki ps)
(100)
400'z
I 9144 mm
; (360")
/ P
"200 I
" 8I
(0.5) (1.0) (1.5)
10 20 30 408) mm (in)
Fig. 5.15 Load versus Deflection at Overhanging End, Ll
-120-
APPENDIX A
FLEXURAL STRESSES IN SINGLE CELL COMPOSITE BOX GIRDERS
A.I Orthotropic Deck and Bottom Flange
The stress function expressions for the top and bottom flanges
are Eqs. 2.26 and 2.28, respectively.
For the flanges, combination of the conditions of symmetry
of Eqso 2.30 with Eqs. 2.19, 2.22, 2.24, 2.26 and 2.28 gives
and
Bn
D = 0n
(A .la)
F = Hn n
o (A .Ib)
By substituting the expression of transverse normal strain as given
by Eq. 2.13g into Eq. 2.31b, performing the integration, and
utilizing Eqs. 2.19, 2.24, 2.26 and A.Ia, the following equation is
obtained.
U = - 1Ex
00 1 12: - [A (- + \Jxzrl)' sinh r 1 O! x1 an n r 1 n
+ Cn (Jl + ~ rZ)sinh r Z a xJr Z xz n sin O! z
n
(A.2)
This equation and the boundary condition of Eq. 2.31a combine to
give the following relationship
c = - C An n n
-121-
(A.3a)
where 2 rl a b(1 + n cr
2\) r
1) sinh
2xz
Cn= (A.3b)
2 r Z O!n br1
(1 + \) crZ ) sinh
2xz
is established.
For the bottom flange, from Eqs. 2.22b, 2.24, 2 0 28 and A.lb
and the boundary condition of Eq. 2.31c, the following relationship
(A.4b)
(A.4a)G - 11 En n n
where r 3 C'i bfcosh n
2lln =
r 4 a bf
coshn
2
For the projecting portions of the concrete deck, the longitud-
inal strain (8 ) at the connection line (x = + b /2) is equal toz - c
that of the deck between two webs. At this line, the transverse
displacement u is zero as is expressed in Eq. 2.31a. Since rigid
body translation and rotation at the deck to web junctions do not2
occur, it follows that (0 ~) = + b /2 = 0, from which the followingoz x - c
boundary condition is deduced by Winter(2.9) •
1E
z
1- - Gzx
( o~ F ) x = bcdX dX 2
(Ao5)
These conditions and Eqs. 2.32 give rise to the following relation-
ships:
-122-
where
t
A == PnA + qn C (A.6a)
n n n
C = s A + t C (A.6b)n n n n n
t
B ==-s A + PnC (A.6c)
n n n n
1
D =- e A + ft C (A.6d)n n n n n
r1
Q' bcosh n c
2Pn 2 2
E2
EZ - _z_)p ](rZ + \> zx) [ r 1( r 1 + \) - -) s + r
Z(r
Z+ \)
zx G n zx G n
1 + zx zx
2 2E
2E
( r 1 + \) zx) [ r 1( r 1 + \)__z_)
t + rZCrZ + \) - _2_) q ]zx G n zx G n
zx zx
(A.6e)
2 + \) br Z r Z Q'zx cosh n c
2 + \) 2f
r1 zx
q =E En
2 2 2(r
Z+ \) zx) [ r 1 ( r 1 + \)
__z_)s + r Z ('r
2+ \) - _z_)p ]
zx G n zx G n1 + zx zx
E E2 2 z
~2 (rZ2 - _z_)q ](r1 + \)zx) [r1(r 1 + \) --) t + + \>zx G n zx G n
zx zx
(Ao 6£)
-123-
sn
2r 1 + \)zx
r 2 + \)2 zx
(A.6g)
r 2 Ct bn c
cosh~2
t =n
(r12 + v )[r1(r1
2 + vE
2E
_z_)t + r (r + \) - _z_)q ]zx zx G n 2 2 zx G n
1 + zx zxE E
2 2 zr2 (rZ
2 - _z_)p ](rZ + \) )[rl(rl + ~ - -)8 + + \)zx zx G n zx G n
zx zx
(A.6h)
1Po Denominator
(A.6i)
r 1 cosh~ 1 Ci (w - b )nee
2 cosh 2
b )c
Denominator
-124-
(A, 6j)
=----------------------sn
r 1 a (w - b ) r Z an(we - be)· h nee sl.·nh -------r l Sln 2 2
Denominator
r 1 a (w - b ) r Z a (w - b )r
2cos~ n ~ c cosh n ~ c
Denomilla tor (A.6k)
r2
tn Denominator
Denominatorr z a (w - b ) r.1 a (w - b )
S ;nh nee cosh neer 1 ~ 2 2
(A. 6 Jl)
r 1 a (w - b )sinh nee- r Z Z
rS P - t sn n n n
rPn
= t t - s qnn n n
rQ = Pn Pn - qn s
n n
r fX = qn t - Pn qnn n
r z a (w - b )cosh nee
2(A.6m)
(A.6n)
(A.60)
CA.6p)
(A.6q)
Equations A.6 relate the stress function coefficients of the
projecting deck to those of the deck between webs. There are also
compatibility conditions between the longitudinal strains of the
flanges and those of the webs along their lines of connection.
Let Z and Z be the total resultant forces of the longitudinalc s
stresses in the concrete deck and bottom flange, respectively.
-125-
w /2Z 2 t f C cr dx
c c0
z
wf
/2z 2 fa N dx
s z
(A.7a)
(A.7b)
Substitution of Eqs. 2.19a, 2.24, 2.26, A.la, and A.6 into A.7a, and
Eqs. 2.22a, 2.24, 2.28 and A.lb into A.7b, respectively, results in
the following.
zc
2 t 2: (~c n=l n
A + ~ . C) sina zn n n n
(A.7c)
zs
where
co
2 ~ (~ • E + Wn . G )n=l n n n
sina zn
(A.7d)
1;1 Q'n bsinh c +2 Pn
r1
O! (w - b )n c c
2
- S r (coshn 1
- Qn rZ
(cosh
r-z O! (w - b )nee
2
r z et (w - b )__n__c__c_ _ 1) J2
--126-
(A.7e)
1r a b r
Zct (w - b )
'f [r sinh ~:2 n c + sinh n c c- qn rln a 2 2 2n
r a (w - b )+ p r (cosh 1 n c c
1)2 -n 1
r O! (w - b )+ t r sinh -2 n c c
n 2 2
r Q'n(wc - b )+rt r (cosh 2 c
1) ]2 -n 2
r3 r
3 CI. bf
wf
- bf 2
r3 Ci b
fsinh n n~n
=- + r3
coshO! 2 2 2n
andr4
O! h f wf
- bf 2 r4 O! b
fr4 nsinh n r
4coshW = + 2n O! 2 2
n
(A.7£)
(A.7g)
(A.7h)
The total internal moment can be considered as composed of1
two parts: a part 2M generated by stresses in the two webs (plus
I Ithe small steel top flanges), and the other part M due to the
longitudinal forces Z and Z. From the equilibrium of the internalc s
moment and the external moment of Eq. 2.29, it is obtained
I 1co
M = ( ~ M sinG' z - Z e + Z e ) (A.8)2 n=l n n s s c c
in which e and e are the absolute values of the distance from thes c
centroid of the web (tee) to the middle line of the bottom flange
and concrete deck, respectively (Fig. 2.8).
-127-
The axial strain of the web at the mid-depth level of concrete
is equal to that of the concrete deck, as given by Eq. 2.13£, along
the line of junction.
Similarly, at the junction of web and bottom flange, with the flange
1E
s
1
- M e c(--II
z + Zc s2 AI)
1E
z(0- - \)z zx
a )x x
bc
2(A.9a)
longitudinal strains given by Eq. 2.16a,
11E
s
}1 1 es
(-r'- -z + z
c s2A' ) =
Es
2 (a3 N( b) za 3 c 3 - 3
- b N)3 x' x
b
2
(A.9b)
In Eqs. A.9, I' is the moment of inertia of the tee, con-
sisting of a web and a small steel top flange, about its horizontal
centroidal axis, and AI the corresponding cross-sectional area.
By substituting Eqs. A.7c, A.7d and A.8 into Eqs. A.9,
expressing the right hand sides of Eqs. A.9 in terms of stress
functions, and making use of the relationsh,ips of stress function
coefficients developed previously, coefficients A and E can ben n
solved and are expressed as shown below.
e It I - e s 11 n1
Ac n M (A. lOa)I , I 1 nn 2E II (9 lln
- , ~ )s n n n
,e Q - e en
Ec n s M (A. lOb)
n , 1 1 n2E II (9 Tin 'n It )
s n n
-128-
2
where
ftn
r 3 Q'n b fcosh
,lln
e ec s(EIJs
2 r 4 <Y b f- Tln r 4 co sh 2n ) (A .10c)
(A .lOd)
Qn
e e1c s (0 - C 'f)(EIt EAT) t c n n n
s s
2e
c 1(EIt + EAT) t (0 -, '±')c n n ns s
+1- r 2) rl an b('V + cosh c
E zx 1 2z
(A.10e)
E1 , (\) + r 2) cosh
n zx 2z
(A .10£)
Once coefficients A and E are determined, all others can ben n
computed from Eqs. A.3, A.4, A.5 and A.6. The stresses can then be
computed- through the stress functions and are summarized as follows:
(1) For concrete deck between webs
crz
,00 ec nn - es lln 2
= ~ (r cosh rt...J ,1 1 I 1 1 an x
n=l ZE II (9 ~ - ( ~ )s n lin ~n n
{ r 2 cosh )M·~n 2· ~2 an x n s~nQ;'n z
-129-"(A. Ila)
f
co e It - e Iln2:
c n s (cosh r0" - - 1 1 1 1O! x
x n=l 2E I (9 Iln 'n ) n- X.s n n
- 'n cosh rZ O! x)M sina zn n n
(A .11b)
00 e it - e Tln2: c n s
(r 1 sinh r: 11" - - f 1 f Q' Xzx n=l 2E I (9 l1 u Cn
n- j{n )s n
- i r2
, sinh rZ
a x)M cosO' z"vn . n n n
(2) For the projecting portions of the deck
(A .1Ie)
1
- e 11s naz
co
~
n=l 2Es
e Itc n1 1
I (9n 11n - ,
nIt )n
f
+ (8n
, 2- 'n t n ) r 2
bh 0: (x - .-£)cos r 2 n 2
b( n + i ) r 2 · h (--£) ]M ·- ~n ~n ft n ·2 Sln r Z an x - 2 n slnan z
(A .lld)
-130-
2 r 3 Q'n b f(r3 cosh 2
(A.l2d)
(4) The projecting portions of the bottom flange are assumed
to be fully effective.
-131-
A.2 Orthotropic Concrete Deck and Isotropic Bottom Flange
All the equations in Section Al which 'are pertinent to the
orthotropic deck remain applicable.
For the isotropic bottom flange, the differential equation of
Eqo 2.21 and the stress function expression of Eq. 2.27 are applicable.
With (a) = (a) = 0, the strains of Eq. 2.16 reduce tox r z r
1E: =
x Es
1 (N - \) N)t
fZ S x
1 (N - \) N)t
fx S z
(A .13a)
(A.13b)
By the symmetry of stress pattern it is obtained
(A .13c)
F = Gn n
o (A.14)
From the assumed boundary condition of Eq. 2.3lc that the transverse
stress is zero at the bottom flange-to-web junctions, it is deduced
H =-11 En n n(A .1Sa)
where
Tin Qln
bf
sinh 2
-132-
(A.ISb)
The small projecting lips of the bottom flange are again
assumed fully effective. The total longitudinal resultant force
in the bottom flange is given by
co
Z = 2 ~ (~ E + W H ) sin Q' zs n=l n n n n n
where
1a b f W - b
f an b fsinh _n__ + f coshl-1n=-
O! 2 2 2n
1 bf(wf - bf) O!n b f
W = [-2 + 4 ] sinh1"1 2
an
+1:-b O:'n b f
(w -£ cosh...._)
2CY f 2n
(A.16a)
(A.16b)
(A.16c)
The compatibility of longitudinal strains at the junction
lines of bottom flange and webs leads to the following:
1E
s
M e(__,_s
I
z + ZC f s)
2A(A.17)
where the right hand side is from Eq. A.13a. After substitution of
the terms into Eq. A.17 and its companion Eq. A.9a, the coefficients
A and E can be solved. The expressions for A and E are exactlyn n n n ,the same as given by Eqs. A.IO except with the values of:K and 11
n n
represented by
2e s I 1
= (------, + ------,)(~ - W ~) + -----Et [coshEI EA n nn sf
s s
2- 11 (-n Q'n
an
bf
bfcosh --- + - sinh
2 2
-133-
(A.18a)
TIne e
(~f 1)(---I jJJnE I E A
s s
(Ao18b)
In all of these equations for orthotropic concrete deck and isotropic
bottom flange, the values of 0 , f and C are from Eqs. A.7e, A.7fn n n
and A.3b,and ~ , w and ~ are from Eqs. A.16b, A.16c and A.ISb.n n n
The stresses in the component parts of the box are summarized
as follows:
(1) For concrete deck between webs:
Same as given by Eqs. A.lla, A.l1b and A.lle.
(2) For the projecting portions of the deck:
Same as given by Eqs. A.lld, A.lle and A.llf.
(3) For bottom flange between webs
N =z
00 e Q t - e r ,c n s ~n
~ 2E 1'(9 ' ~ , - r t ')n=l s n lin ~n ft n
[cosh a x - ~ (~ coshn lin Ci
nct x + x sinh
n
M sin Q:' Zn n
00 e 9 • - e , •N = - l: ens n
x 2E I 1 (9 t "n ' _ r' I,n=l s n lin ~n f{n)
[cosh Ci x - 11 x sinh Q' x] M sin O! zn n n n n
-134-
(A.19a)
(A. 19b)
00 e 9 , - e r'c n s ~n
2: 2E I I (9 ' 'Il ' - , ' 1'(. ') •n=l s n n n n
Nzx
[sinh Q' xn
sinh O! x + x cosh. O! x) ] ·n n
M coshQ! Zn n (A.19c)
(4) For the projecting lips of bottom flange:
Nz
[cosh
+ :f sinh Q'n2
bf)] . M
nsipa z
n(A.19d)
(Assumed to be uniform from juncture to tip.)
N 0 (Assumed)x
OJ e 9 t - e~ 'nI
N i:c n
CY- - I' (9 I I
'nI ft, I)
.zx 2E 11n - n
n=l s n n
(A.1ge)
an b f[cosh -2-2 CLn b f
l1n
(a- ,cosh, 2n
b f Ci b f W f+ - s"inh n ) ] (- _ x) M co s2 2 2 n Q'n Z
(A.19f)
(5) For a web:
The moment, shear and axial force taken by a web and a
small steel top flange are the same as given by Eqs. A.l1m,
A.lln and A.llo. The normal stress, cr , is given by Eq. A.llp.z
The shearing stress is-135-
co
a =x
e c ~n - e s ~n I
~ ---.;;...,---:'I~-'---:::'------'- [(Pnn=l 2E I (9 ~ -; ~ )s n lin ~n n
bh ( ~)cos r 1 Q'n x - 2
b- (Sn + ~n Pn) sinh r 1 an(x - 2
C)
I
+ (sn
,- C t )n n cosh
b- (9 + C K) sinh r a (x -£)JM si~ z
n nn 2n 2 n n
1'"zx
co
l.:n=l 2E
s
e ft - ec n s
,lln
I I
i It )~n n
(A. lIe)
- (S + ~ p) r 1n n n
b( ~)cosh r 1 an x - 2
, , bc+ (9 - , t ) r Z sinh T Z O!n(x - -)
n n n 2
b- (Q +' K) r Z cosh r Z Cin(X - .-£) ]M co sa z
n n n 2 n n
(A. 11~)
-136-
(3) For the bottom flange between webs
OJ: e 9 - e 'n 2N 2:
c n s (r3
cosh r3
::::: , Q:' Xz If (9
, r f n0=1 2E n l1n - C K )s n n
11n2
cosh x)M sin Ci (A .llg)r4
r 4 an zn n
,co e 9 - e Cn
N ~c n s ,(cosh= - , , I r
3O! x
x, n
n=1 2E I (8 I1 n - Cn ftn )s n
TIn cosh r 4Ql
nx)M sin Q:' z (A.llh)
n n
f
co e 8 - e 'nN :::: - L:c n s (r
3sinh r
3r f I f ex xzx 1 0
0='1 2E I (8 11 - Cn ft n )s n n
- TI r 4 sinh r 4 Ci x)M cos a z (A.IIi)n n n n
(4) For the projecting portion of the bottom flange
Nz
=OJ
L:n=1 2E
s
ec
t
I
1Q - e ,
n s n
2 r 4 Q' b f- rn r cosh n ) M sin'In 4 2 n Q:'n Z
(Assumed to be uniform from junction to tip) (A.llj)
Nx
o (Assumed)
-137-
(A.l1k)
,f{ )
n
,- ~n
C fn9 - e
n se
ci:
n=l 2Es
Nzx
- 11n
(A. l1.e)
(5) For a web (plus the small steel top flange)
co
M' = ~ L; [1n=1
,es(~n - Wn I1n)(ec gn - e s,
ft )n
,t e (0 - ~ I )(e ~ - e ~n)+ c c n n ~n c '~n s II ]------1--'-----1----- M
Es I (9 ~n - 'n ~n) nsin Q'
nz (A. 11m)
, I co
V = - ~ [1 2 n=l
,e (l-L - W 11 ) (e 8 - es n nne n s
1 '1 'IEs I (Qn ~n - 'n ~n)
O! Mn n
cos Q'n
z
(A. lIn)
,N
-1
2E Is
00 ,
~ t (0 - ~ C)(e ~t..J [c n nne 'n
n=1 ,.Qn I1 n - Cn ~n
+
, f
(~ - w 11 )(e Q - e Cn )__..;..;;;.n__n~_n__c:__-n~--s---JM sin Q' Z
, 'f n nQn Tin - 'n f't n
, ,-R-+1LLcr - f t
Z A I
(A.llc)
(A. IIp)
-138-
in which yt is the vertical distance from the centroid of the
web (tee) to the fiber considered, being positive if downward.
,1
co e Q - e s Cnr 3 O! b
fl:
c n ,[r3
nT = , I , sinh
2zy2E I t n=l 9 'fln - ~n ft
s w n n
r 4 a bf w - b
f 2r
3Of b
f- 11
n fO!n(r3
cosh nr 4 sinh2 2 2n
b ' ,~ 2 cosh r 4 an f)JM cos a z +~lin r 4 2 n n
I tw
(dN ) A+ dz 1
A tw
(A.llq)
where Q is the statical moment of area (AI) about the horizon
tal centroidal axis of the web (tee)(x' - axis in Fig. 2.8),
Al is taken from the section in consideration to the bottomI
dNoutermost fiber of the web (tee) and dz is the derivative
,of N with respect to z. The individual equivalent widths of
the flange portions are computed below.
(1) For concrete deck between webs:
By definition, the equivalent width, ~l (Fig. 2.9) can be
found by
tc
(0- )z X
bc
2
b /2= J c
o
-139-
0'z
tc
dx (A .12a)
By substituting Eq. A.Ila into the above, integrating and nondimen-
sionalizing with b , it is obtainedc
(2) For the projecting portion of the deck:
wc - b c
2~ { ro-TT-Cw-"';"";;'--b-)- ~
c c n=1
I
ec!'tn - e s l1 n--I--l--~I-~I-[ CO!(8n ~n - en ~n)n n
(£1n
(A.12c)
r1
O! bn c)- r 1 sinh
2
t~le ft - e llnC n s
f ,Q Tin - C ft
n n n
M sin an z}n
r z Q!n b~ Ca Y - r sinh c)JM sinn n n 2 2 n O!n
-140-
2Es "'
-, f'tn n
- e s ~n8n
ec
co
2:n=l 9
n
1= -----
l ' tw
'fzy
· an
~ b f 2 an b f[cosh~ - ~ (-- cosh ---2
n ~n
b b I , (dN')Af Q'n f + L.JL + dz 1+ -Z sinh ----Z-)]}Mn cos an Z 1 1 t At t
W w
(A.19g)
where Q1 and Al are determined by the same procedure as that
for Eq. A.Ilq.
The equivalent widths are the following:
(1) For concrete deck between webs:
Same as expressed by Eq. A.12b.
(a) For the projecting portion of the deck:
Same as expressed by Eq. A.12c.
(3) For bottom flange between webs:
2i {co e c Qn - e 'n ' Cl:'n b f= -- 'E 1 I S I I [sinh --2-
TT b f n=l (9 ;)n TIn \:In Xn n
1Q' b
fb
f an h f 1/- 11 (- sinh _n__ +- cosh ---2-) ]Mn
sin an O! 2 2 n
n
{n~l1
e Q - e CnO! b
f 2 Q'n b fc n sf 1 [cosh~ - 11 (- cosh --2-
Q Tin - C ftnn Q'n
n n
(A.20)
-141~
(4) The projecting portions of the bottom flange are assumed to
be fully effective.
A.3 Isotropic Concrete Deck and Bottom Flange
Equations of Section A.2 which are derived for an isotropic
bottom flange are applicable. These are Eqs. A.14, A.lS, A.16 and
A.18. For an isotropic deck
and
Ez
\)zx
Ex
\)
xz
(A.21a)
(A.21b)
The governing differential equation is Eq. 2.21 and the stress
function expressions are Eqs. 2.24 and 2.25. All the procedures in
determining the stress function coefficients are the same as those
employed previously. The results are listed in the following:
sinhCt bn c
2
bc
2cosh
Q' bn c
2
1 - \)xz---
I + \)'xz
1Q'
nsinh
Ci bn c
2
(A.22)
P I =n
21 + O! (1 + \) )
n zx
Ci bcosh,. --lL-.£
2E E
Ot. (1 + v z(3 + z- -)8 - 'V - -)pn zx G n zx G n
zx zxE E
O! (1 +v _z_)t (3 + z- V G)qnn zx G n zxzx zx
(A.23a)
-142-
2 a b b Q' b
O! (1 + \'zx)cosh~+~ sinh~
n2 2 2
qn =E E
O! (1 + z(3 + z
\) - -)8 - \) - -)p2
n zx G n zx G n1 + zx zx
O! (1 + \)zx) E En __2_) tex (1 + \) - (3 + \) _Z_)q
n zx G n 'zx G nzx zx
(A.23b)
O! (1 + \)zx) a bn cosh n c
S ::::2 2
n E EO! (1 + z z
a (1 + \) zx)\> - -)t - (3 + \) - -)q
n zx G n zx G n
1 + n zx zx2 E E
a (1 + z(3 + z
\) - -)8 - \) C;-)Pnn zx G n zxzx zx
(A.23c)
E EOf (1 + \) - _z_) t ~ (3 + \) __2_) q
n 'zx G n 'zx G nzx zxE E
O! (1 + \) z ) ( zn zx - C-- sn - 3 + v ---G)Pnzx zx zx
tn
O!n1 +
Of b O! (1 + \) )cosh~ + n 'zx
2 2
(1 + \) )zx .
2
bc2 sinh
Ci bn c
2
1Pn Denominator
(A.23d)
(A.23e)
1 - 2 CL (w - b )sinh nee
O! 2n
qn Denominator(A.23f)
sn
1O! .
n
2cosh
Denominator
-143-
(A.23g)
Denominator2
tn
w( c
- b 2c )
(A.23h)
w - b 1 a (w - b ) QI (w - b )Denominator
c c sinh n c c cosh n c c2 Q' 2 2
n
(A.23i)
,Sn t s - s Pn
CA.23j)n n n
r CA. 23k)Pns qn t tn n n
rQ Pn Pn qn s (A.23£)
n n
Ii qn t - Pn qn CA. 23m)n n
1Q' b
1a (w -b ) O! (w -b )
~ (sinh nZ
c + nee [ n c c=- Pnsinh - Sn cosh 2 - 1Jn O! 2
n
1 [ 1O! (w - b ) w -b Q' (w -b )
+ sinh nee+~ cosh nee]
S -2n O! 2 2
n
9 [l Ccoshn a
n
O! (w -b ) w -b O! (w -b )nee _ 1) +~ sinh n c c J}
2 22-
(A.24a)
a b b a b11. ne c ne'1'n = ;-( <;- smh Z +""2 cosh -2-) + qn
n n
Q' (w· h n c
S l:n 2b )
c
+ P [coshn
w -b+ .-E-..£
2
O! (w -b ) Q' (w -b )nee _ 1J + t I [~sinh nee
2 n O! 2n
O! (w -b ) 1 O! (w -b )cosh n ~ c ] + It [-(cosh nee _ 1)
. n Q' 2n
w -b Q' (w -b )+~ sinh n ~ c]}
-144-
(A.24b)
The expressions for ~ , ~ , and ware identical to Eqs. A.ISb, A.16b,n n n
and A.16c, respectively.
9n
e ec s
(El's
(A.25a)
e 2 1 + v a bc 1 zx n C(El' + EAT) t c (0n - Cn '±'n) + -E-- cosh -2-
s s z
a b b a b, 1-[.1... cosh~ + c (1 + \J ) sinh ~2c Jn E O! 2 2 zx
z n(A.25b)
The expressions for uland ~ 1 are the same as Eqs. A.18a and A.18b.n n
The following is a summary of expressions for stresses:
(1) For the deck between webs:
~ 2E I f (g 1'r1 ' _ I 1 ')n-1 s n tin ~n ft n
crz
=OJ e ft I - e T\ 1
ens n[cosh O!
nx
crx
1" --zx
r (1... cosh O! -f~ x sinh O! x)]M sin O!'on O! n n n n
n
co e It ' - e T\ 1
~ ens n (cosh a x_ 2E I' (9 '''n 1 - i 1 ft ~ nn-1 s n lIn ~n n
-, xsinha ',x)M sinet Zn n n n
co e ft. 1 - e 11 1
C n s n [~ sinh O! .
nn=l 2E 1 1 (9 tTl ' - , I !'t. ')
s n n n n
z (A.26a)
(A.26b)
1-, (- sinh Q! x+x cosh Q' x)J M cos a z
na n n n nn
-145-
(A.26c)
(2) For the portions of the deck outside webs:
(j =z
co e ~ , - e T! ' b~ ens n r[p , _ C q ')cosh a (x _ ~)
2E r I (9 r "n til '~n n n n 2n=l s n lin - ~n Kn )
b b-(S +C p ) sinh ct (x - --.£)-(g +~ tt )[-L sinh a (x - .-.£.)
n n n n 2 n n nan 2n
b b b+(x - c) h a (x - --2c)] + (sn' - r t ,)[-L cosh a (x -~)2 cos rl ~n nan 2
n
b b+(x - 2
c)sinh an(x - 2
c)J} Mn sin a z
n(A. 26d)
ax
co e ft I - e 11 ' b~ ens n [(p , _ C q I )cosh ct (x - Zc)
n=12E rl(e f 11 1 _ , f ~ ') n n n ns n n n n
b b+(8 '_~ t ')(x - 2c)sinh a (x - ~)JM sin a z
nn n n 2 n n (A.26e)
'rZX
- -
OJ e ft • - e 11 tens n [.
~ (pn= 1 2E I ' (e 1 11 • - , 'ft. ') n
s n n n n
b, q ')sinh a (x - 2c)
n n n
b 1 b-(S +' p )cosh a (x - ~)-(g +' ft ) [- cosh ct (x - --£)n n n n 2 n n n O! n 2
n
b b+ (x - :f) sinh an(x - ;)]
b+ (s '-C t ') [.1- sinh a (x - --£). n n n O! n 2
n
b b+ (x - ;f) cosh an(x - ;)]}Mn
-146-
cos Ct zn
(A.26f)
(3) For the bottom flange between webs:
Same as given by Eqs. A.19a, h.19b and A.19c.
(4) For the projecting lips of bottom flange:
Same as given by Eqs. A.19d, A.lge and A.19f.
(5) For the webs:
The expressions for moment, shear and axial force taken
by a web and a small steel top flange are the same as
Eqs. A.llm, A.lln and Aollo, respectively.
The normal stress is computed from Eq. A.lIp and the
shearing stress from Eq. A.19g.
The equivalent widths are summarized as follows:
(1) For the deck between webs:
Al = --.l:.L {~ __e_c_ft_n__-_e_s_ll_n__b 11 b ttl
C c n=l (Qn ~n en ~n)n
QI b[ · h n cS1.n --2-
b O! b 1 Q' b- en (2
ccosh~ + ;- sinh ~) ]Mn sin an
n
- es
- Cn
Tl n ,rt .
n
a b[ n ccosh --2-
2Q' b
n cen (;- co sh -2-n
b Q' b }+ -!f sinh T) ]Mn sin an Z
-147-
CA.27a)
(2) For the projecting portion of the deck:
f {A2 Z,e
co e ft - e flu Q. bc n s
f/J · h n c)- b n(w - b ) L: I I I 1 (an Sln --2-
w n=l (8 - , nc c c c fln ft ) n
n n n
-{- (s f - , t ') L JM sin O'n J.n n n O'n n Jr
CA.27b)
(3) For the bottom flange between webs:
Same as given by Eq. A.20.
(4) The projecting portions of the bottom flange are assumed
to be fully effective.
-148-
APPENDIX B
ROTATIONS AND DERIVATIVES OF WARPING FUNCTION
B.1 Simple Beam with Concentrated Torsional Moment (Fig. 3.5a)
a :s; Z :s; aL
TL ~ ~o= Gr
KT
[ A L (sinh a AL ctnh L - cosh a AL)sinh AZ
z+ (1 - a) - ]L
fft = _T_ \(sinh a AL ctnh I\L - cosh O! I\L) sinh AZG
rK
T
(B .la)
(B. 1b)
Section torque (1 - oD T (B .lc)
Q'L :s; Z s: L
o =~ [:L sinh a AL (ctnh AL sinh AZ - cosh AZ)G
rK
Tz+ OL (1 - L) ] (B .ld)
ftl = _T_ A sinh Of AL (ctnh AL sinh \Z - cosh AZ)G
rK
T(B. Ie)
Section torque - OL T
-149-
(B .If)
B.2 Concentrated Torsional Moment at Overhanging End of
Simple Beam (Fig. 3.5b)
o ::;; Z ;5; L
1-.L T L 1o ~ ------ · n [(sinh AL ctnh ALl - cosh AL) sinh AZG
rK
T
+ sinh AL (cosh AL - sinh AL ctnh Z (B.2a)ALI) L ]
TL ;.,..2sinh AL ctnh i\L - cosh AL
fll 1sinh AZ (B.2b)= ----
Gr K TD
Section torque
.- ~ sinh i\L (sinh AL ctnh i\LI - cosh i\L). T (B.2c)
D
o= GTLK · -n1 [AL - ~ sinh i\L (cosh AZr T
- ctnh i\LI sinh i\Z)] +~G
rK
T(B.2d)
f" = TLK i\2 • sinh At (ctnh i\LI
sinh i\Z - cosh I\.Z)Gr T D
Section torque T
where
D =-i\L + ~ sinh i\L (cosh AL - sinh AL ctnh ALI)
-150-
(B.2e)
(B.2f)
(B.2g)
B,3 Simple Beam with Uniformly Distributed Torsional Moment in
Part of Span (Fig, 3.5c)
{cosh AZ - [sinh a ~L
+ (1 - cosh a ~L) ctnh AL] sinh ~Z - 1J
aL fit Z+ 2G
rK
T(2 - a - aL) Z
ill
£" = _t_ [cosh \Z - [sinh Q' ALG
rKr
+ (1 - cosh O! A.L) ctnh I\L] .sinh I\Z - I}
(B.3a)
(Bo 3b)
Sect{~n torque (B.3c)
Q'L ~ Z ~ L
a2L
2- ctnh I\L sinh AZ) + ~---
2Gr
KT
(B,3d)
rotf" = -- (1 - cosh Q' i\.L) (cosh I\Z - ctnh AL sinh AZ)
Gr
KT
(B.3e)
Section torquea2 L
- -2- 'mt
(B,3£)
In the case where IDt
extends over the whole span, Eqs. B.3a
through B.3c can be used by setting a equal to 1.
-151-·
B.4 Uniformly Distributed Torsional Moment Throughout Overhanging
Portion of Simple Beam (Fig. 3.5d)
o ~ Z ~ L
C/J =
A(L1 - L) sinh A(LI - L) - cosh A(LI - L) + 1
D
(L sinh AZ - Z sinh AL)
rotff! = -- · AL • [ACL - L) •
Gr
KT
D 1
(B.4a)
sinh A(LI - L) - cosh A(L1 - L) + 1J sinh AZ
(B.4b)
sinh i\LSection torque = - ~ rot' AD
[ACLI - L) sinh A(LI - L) - cosh A(LI - L) + 1J
(B.4c)
- ~J sinh AL [sinh A(11 - Z) - sinh A(11 - L)]
+ AL cosh AL [sinh A(LI - Z) - sinh A(Ll - L)J
+ ~ sinh AL sinh ACL - Z) + AL(sinh AZ - sinh AL)}
+ GffitK (Z _ L)(L1
_ Z ; L)r T
-152-
(B.4d)
_~ 1 2fff D ([i\ L(L1 - L) - ~J sinh i\L sinh i\(L
1- Z)- G
rIZ
T
+ i\L cosh i\L sinh i\(L1
- Z) + ~ sinh i\L sinh i\(L - Z)
where
fit+ i\L sinh i\Z} - -
Gr
KT
Section torque ; (LI
- Z) fit
(B.4e)
(B.4£)
D = sinh i\L1[i\L + ~ sinh i\L(sinh i\L ctnh ALI - cosh i\L)]
(B.4g)
-153-
A
Af
A--m
~ to HN
A0
*A
a, b, c
(ax)r' (a )z r
B
bc
bf
C1
to Cs
d
df
E
Ef
Em
Er
Es
NOTATIONS
Area
Ratio of reinforcing fiber area to cross-sectional
area of composite
Ratio of matrix area to cross-sectional area of
composite
Coefficients in expression for Xn
Enclosed area of box section
Transformed area of cross section
Coefficients of differential equation of stress
function for concrete deck
Cross-sectional area of bottom flange transverse
sti££ener(x) and longitudinal stiffener (z)
Bimoment (due to warping)
Concrete deck width between webs of box girder
Bottom flange width between webs of box girder
Coefficients in the equation of angle of twist, 0,
and warping function, f
Diameter
Diameter of fiber
Young's modulus
Young's modulus of reinforcing fiber in composite
Young's modulus of matrix of composite
Young's modulus of reference material
Young's modulus of steel
-154-
Et
F
fez)
G
hw
I
Ic
Is
I , I , Ix Y z
I xy
Iw
1_w
I- , I-wx wy
KT
L
M
Mn
~
M, M, Mx y z
NOTATIONS (continued)
Equivalent modulus of elasticity of reinforced
concrete deck in the longitudinal (z) direction
Airy's stress function
Warping function
Shear modulus; G's with subscript correspond to E's
Height of web
Moment of inertia
Central moment of inertia
Moment of inertia of flange with respect to neutral
axis of box
Moment of inertia with respect to x, y and z axis,
respectively
Product moment of inertia
Moment of inertia of web with respect to neutral
axis of box
Warping moment of inertia
Warping moment of inertia with respect to x and y
axis, respectively
St. Venant torsional constant
Span length
Bending moment
Coefficient (moment) in sine series expression of
moment
Torque
Bending moment with respect to x, y, z axis,
respectively
-155-
mt
mz
N
Nx' Nz
Nzx
P
p, q, r
Q
Q
q
rn
s
sws
s x
sz
NOTATION (continued)
Uniformly distributed torsional moment
Distributed torsional moment
Forces per unit length of plate
Axial force in x and z direction, respectively
Shear force in zx plane
Load
Coefficient of differential equation of stress
function for bottom flange
Static moment of area of cut cross-section
Adjusted state moment of area of cut cross-section
Shear flow
Shear flow in cut cross-section
Shear flow to impose compatibility at cut of
cross-section
St. Venant shear flow
Coefficient in expression for Xn
Element of matrix relating stress to strain, with
subsc;.ripts
Warping static moment
Distance along wall of box section; spacing of
stiffeners
Spacing of longitudinal stiffener in x direction
of bottom flange
Spacing of transverse stiffener in z direction of
bottom flange
-156-
t
tc
t.1
u
us
v
w
Wb
wc
w0
wtf
Xn
x, y, z
Zc' Zs
a,n
y
8
€
NOTATION (continued)
Thickness
Thickness of concrete deck
Thickness of bottom flange
Thickness of element i
Thickness of steel top flange
Displacement along x-direction
Displacement in the tangential direction of
cross-section
Shear force
Width; displacement along z-direction (axial
displacement)
Overall width of bottom flange
Overall width of concrete deck
Axial displacement at s = a
Width of steel top flange
Function of x in Fourier's series
Centroidal principal axes
Total axial force in concrete deck and bottom
(steel) flange, repectively
Coefficient in Fourier's series
Shearing strain
Deflection
Normal strain
-157-
Al
A2
A3
V
v
v12
Vf
vm
Vs
P
Po
a
°B
an
~
a , a , ax y z
(0z)p
(0z)r
T
TB
NOTATION (continued)
Nondimensional coefficient in warping function fez)
and angle of twist 0(z); effective width
Effective width of concrete deck between webs
Effective width of overhanging portion of concrete
deck
Effective width of bottom flange between webs
Warping shear parameter
Poisson's ratio
Poisson's ratio of composite in direction 1 due to
stresses applied in direction 2
Poisson's ratio of reinforcing fiber of composite
Poisson's ratio of matrix of composite
Poisson's ratio of steel
Distance from centroid to tangent on cell wall
Distance from shear center to tangent on cell wall
Normal stress
Bending normal stress
Distortional normal stress
Warping normal stress
Stress along x, y, z direction, respectively
Longitudinal stress in the plate of the bottom flange
Longitudinal stress in the reinforcing stiffeners
of the bottom flange
Shear stress
Flexural shearing stress
-158~
¢(z)
¢'
w
wo
Wn
NOTATION (continued)
Distortional shearing stress
Warping tortional shearing stress
St. Venant shear stress
Angle of twist
Angle of twist per unit length
Unit warping with respect to centroid
Unit warping function with respect to shear center
Normalized unit warping with respect to shear center
-159-
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STEEL BOX GIRDER BRIDGES ULTIMATE STRENGTH CONSIDERATIONS,Journal of the Structural Division, ASCE, Vol. 100, No.8T12, Proc. Paper 11014, December 1974.
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TRENDS IN THE DESIGN OF BOX GIRDER BRIDGES, Journal of theStructural Division, ASeE, Vol. 93, No. ST3, Proe. Paper5278, June 1967.
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-160-
REFERENCES (continued)
1.9 V1asov, V. Z.,THIN-WALLED ELASTIC BEAMS, The Israel Program for ScientificTranslation, Jerusalem" 1961; Available from the Office ofTechnical Service~ U. S. Department of Commerce, Washington,D. C.
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1.11 Abdel-Samed, S. R.} Wright, R. N. and Robinson, A. R~,
ANALYSIS OF BOX GIRDERS WITH DIAPHRAGMS, Journal of theStructural Division, ASeE} Vol. 94, No. STIO, Proe. Paper6153, October 1968.
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1.14 Lim, P. T. K. and Moffat, K. R.,GENERAL PURPOSE FINITE ELEMENT PROGRAM, Proe. Symposium onBridge Program Review, P.T~R~C., London, 1971.
1.15 Cheung, Y. K.,ANALYSIS OF BOX GIRDER BRIDGES BY FINITE STRIP METHOD,Paper presented at 2nd International Symposium on ConcreteBridge Design, Chicago, March 1969.
2.1 Rive11o, R. M.,THEORY AND ANALYSIS OF FLIGHT STRUCTURES) McGraw-Hill BookCompany, Inc., New York, 1969.
2.2 Garg, S. K., Svalbonas, V. and Gurtman, G. A.,ANALYSIS OF STRUCTURAL COMPOSITE MATERIALS, Marcel Dekker,Inc., New York, N.Y.) 1973.
2.3 Ekvall, J. C.,ELASTIC PROPERTIES OF ORTHOTROPIC MONOFILAMENT LAMINATES~
ASME Paper No. 6l-AV-56, presented at the ASME AviationConference, March 1961.
-161-
REFERENCES (continued)
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1948.
2.8 Flint, A. R. and Horne, M. R~,
CONCLUSIONS OF RESEARCH PROGRAMS AND SUMMARY OF PARAMETRICSTUDIES, Proe. of the International Conference organized byThe Institution of Civil Engineers in London, 13-14,February 1973.
2.9 Winter, G.,STRESS DISTRIBUTION IN AND EQUIVALENT WIDTHS OF FLANGES OFWIDE, THIN-WALL STEEL BEAMS} NACA Technical Note No. 784,November 1940.
2.10 Tate, M. B.,SHEAR LAG IN TENSION PANELS AND BOX BEAMS, Iowa EngineeringExperiment Station, Iowa State College, Engineering ReportNo.3, 1950.
2.11 Hildebrand, F. B. and Reissner, E.,LEAST-WORK ANALYSIS OF THE PROBLEM OF SHEAR LAG IN BOXBEAMS, NACA Technical Note No. 893, May 1943.
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-162-
REFERENCES (continued)
2.13 Lim, P. T. K., .ELASTIC ANALYSIS OF BRIDGE STRUCTURES BY THE FINITEELEMENT METHOD, Ph. D. Dissertation, University ofLondon.
2.14 AASHTO,STANDARD SPECIFICATIONS FOR HIGHWAY BRIDGES, The AmericanAssociation of State Highway and Transportation Officials,General Offices, 341 National Press Building, Bashington,D.C., 11th Edition, 1973.
2.15 Malcom, D. J. and Redwood, R. G.,SHEAR LAG IN STIFFENED BOX GIRDERS, Journal of theStructural Division, ASCE, Vol. 96, No. ST7, July 1970.
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3.2 Galambos, T. V.,STRUCTURAL MEMBERS AND FRAMES, Prentice-Hall, Inc.,Englewood Cliffs, N. J., 1968.
3.3 Dabrowski, R.,CURVED THIN-WALLED GIRDERS~ THEORY AND ANALYSIS, Translatedfrom the German by C. V. Amerongen, Cement and ConcreteAssociation, Translation No. 144, London, 1972.
3.4 McManus, Paul F. and Culver, Charles G.,NONUNIFORM TORSION OF COMPOSITE BEAMS, Journal of theStructural Division, Prac. of the ASCE,- ST6, June 1969.
3.5 Kuo, J. T. C. and Heins, C. P.,BEHAVIOR OF COMPOSITE BEAMS SUBJECTED TO TORSION, ReportNo. 39, Department of Civil Engineering, University ofMaryland, February 1971.
3.6 Heins, C. P. and Kuo, Je T. C.,COMPOSITE BEAMS IN TORSION, Journal of the StructuralDivision, Froe. of the ASeE) STS, May 1972.
3.7 Benscoter, S. U.,A THEORY OF TORSION BENDING FOR MULTICELL BEAMS, Journalof Applied Mechanics l ASEM~ March 1954p
-163-
REFERENCES (continued)
3.8 Conway, H. D., Chow, L. and Morgan, G. W.,ANALYSIS OF DEEP BEAMS, Journal of Applied Mechanics,ASME, June 1951~
3.9 Yen, B. T. and Chen, Y. S.,TESTING OF LARGE SIZE COMPOSITE BOX GIRDERS, FritzEngineering Laboratory Report No. 380.11 (Draft)Lehigh University, Bethlehem~ PA (November 1976)
4.1 Dalton, D. C. and Richmond, B.)TWISTING OF THIN-WALLED BOX GIRDERS OF TRAPEZOIDAL CROSSSECTION, Proe. Institution of Civil Engineers, Vol. 39,January 1968.
4.2 Sakai, F. and Okumura) T.,INFLUENCE OF DIAPHRAGMS ON BEHAVIOR OF BOX GIRDERS WITHDEFORMABLE CROSS SECTION} Preliminary Report of IABSE 9thCongress, Amsterdam, May 1972.
4.3 Chapman, J. C., Dowling, P. J., Lim j P. T. K. andBillington, C. J.,
THE STRUCTURAL BEHAVIOR OF STEEL AND CONCRETE BOX GIRDERBRIDGES, Vol. 49, No.3, The Structural Engineer, March1971.
4.4 Yen, B. T., Hall, J. and Chen} Y. S.,AFFECT OF DIAPHRAGM SPACING ON BOX GIRDERS, FritzEngineering Laboratory Report No. 380.13 (in preparation)Department of Civil Engineering, Lehigh University,Bethlehem, PA.
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5.2 Eppes, B. G.,COMPARISON OF MEASURED AND CALCULATED STIFFNESS FOR BEAMSREINFORCED IN TENSION ONLY, Journal of the ACT} Voll 31~
No.4, October 1959.
5.3 Yu, W. W. and Winter, G.,INSTANTANEOUS AND LONG-TIME DEFLECTIONS OF REINFORCEDCONCRETE BEAMS UNDER WORKING LOADS, Journal of the ACI~
Vol. 32, No.1, July 1960.-164-